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Ann. Scient. Éc. Norm. Sup.,
4e série, t. 36, 2003, p. 479 à 524.
DIAGRAM ALGEBRAS, HECKE ALGEBRAS AND
DECOMPOSITION NUMBERS AT ROOTS OF UNITY
B Y J.J. GRAHAM
AND
G.I. LEHRER
A BSTRACT . – We prove that the cell modules of the affine Temperley–Lieb algebra have the same
composition factors, when regarded as modules for the affine Hecke algebra of type A, as certain
standard modules which are defined homologically. En route, we relate these to the cell modules of the
Temperley–Lieb algebra of type B, which provides a connection between Temperley–Lieb algebras on n
and n − 1 strings. Applications include the explicit determination of some decomposition numbers of
standard modules at roots of unity, which in turn has implications for certain Kazhdan–Lusztig polynomials
associated with nilpotent orbit closures. The methods involve the study of the relationships among
several algebras defined by concatenation of braid-like diagrams and between these and Hecke algebras.
Connections are made with earlier work of Bernstein–Zelevinsky on the “generic case” and of Jones on
link invariants.
 2003 Éditions scientifiques et médicales Elsevier SAS
R ÉSUMÉ . – Nous démontrons que les “modules cellulaires” de l’algèbre de Temperley–Lieb affine ont,
regardés comme modules pour l’algèbre de Hecke affine de type A, les mêmes facteurs de composition que
certains modules “standards” qui sont définis homologiquement. Au passage, nous relions ces modules aux
modules cellulaires pour l’algèbre de Temperley–Lieb de type B. Parmi les applications est la détermination
explicite des nombres de décomposition de certains modules standards aux racines de l’unité, qui implique
à son tour la détermination de certains polynômes de Kazhdan–Lusztig associés aux clôtures d’orbites
nilpotentes. Nos méthodes consistent à étudier les rapports entre certaines algèbres de concaténation de
diagrammes de tresses ou analogues, et entre ces algèbres et les algèbres de Hecke. Le travail est aussi relié
aux travaux précédents de Bernstein–Zelevinski dans le cas “génerique” et de Jones sur les invariants des
entrelacs.
 2003 Éditions scientifiques et médicales Elsevier SAS
1. Introduction
Let R be a commutative ring with 1 and let q ∈ R× , where for any ring A with 1, A× denotes
the group of its invertible elements. In [14], we defined a “Temperley–Lieb category” Ta , whose
objects are the natural numbers N, and whose morphisms are R-linear combinations of “planar
diagrams from t to n” (for t, n ∈ N), with composition depending on the element q. In particular,
we have the algebras of endomorphisms
Tna (q) := HomTa (n, n),
for n = 0, 1, 2, 3, . . ..
These were called in [14] the affine Temperley–Lieb algebras. Using a calculus of diagrams,
together with the philosophy of cellular algebras, we developed in [op. cit.] a theory of cell
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J.J. GRAHAM AND G.I. LEHRER
modules Wt,z (n) (0 t n, n − t ∈ 2Z, z ∈ R× ) for the algebras Tna (q), and gave a complete
description of the composition factors of the cell modules, valid for all q, even for q a root of
unity, the most complicated case. The description of the irreducible Tna (q)-modules is valid when
R is any algebraically closed field. The analysis of the composition factors of the cell modules
applies when R is an algebraically closed field whose characteristic is either 0 or p > 0 such that
pe > n, where e is the multiplicative order of q 2 .
na (q) be the extended affine Hecke algebra of type A
Now let H
n−1 , which corresponds to
G = GLn (C). This is the algebra considered in [4,28,19], for which there is a theory of “standard
na (q) as a convolution algebra of
modules” Ms,N , which may be constructed by regarding H
coherent sheaves acting on the Borel–Moore homology of certain varieties Vs,N , where s is a
semisimple element of G and N is a nilpotent element of G = Lie(G) such that Ad(s)N = q 2 N .
When q is not a root of unity, the structure of the modules Ms,N is fairly well understood, while
when q is a root of unity, it was conjectured in [20] and proved in [1] (cf. also [10]), that the
decomposition numbers of the standard modules, i.e. the multiplicities of the irreducibles in
Ms,N , are given by values of certain Kazhdan–Lusztig polynomials, which are generally not
known explicitly (see also [21]).
In this work, we show that the cell modules Wt,z (n) of the algebra Tna (q) (cf. [14]) may
na (q) → Tna (q) (see (5.12) below) to
be inflated via a family {ψα | α ∈ R× } of surjections H
a (q), which we identify explicitly (in the Grothendieck group Γ(H
a (q)) of finite
modules for H
n
n
a
dimensional Hn (q)-modules) with the standard modules Ms,N where N has just two Jordan
blocks. This enables us to use the results of [14] to give completely explicit decompositions of
these standard modules, and therefore give character formulae for their irreducible heads. We
also obtain much detailed information about their internal structure. Among the consequences
of our results are the statements that Ms,N is always multiplicity free, and that when q is a root
of unity, the composition length may be arbitrarily large (as n increases). The key point in this
work is the identification of the inflations of our cell modules with the standard modules up to
Grothendieck equivalence (Theorem (9.8)).
To achieve this, we show that the inflations ψα∗ Wt,z (n) are generically (i.e. for generic q)
na (q). For this we need to understand the
induced modules from a parabolic subalgebra of H
action of the “translation elements” Xi on the inflations. This in turn depends on the relationship
na (q); the first as a twisted tensor product of the
between two (known) ways of viewing H
∼
group ring R[V ] of Z = V with the Hecke algebra Hna (q) of the Coxeter system of affine
type A
n−1 , the second as the tensor product of the finite dimensional Hecke algebra Hn (q)
of type An−1 with the R-algebra of Laurent polynomials R[X1±1 , . . . , Xn±1 ]. We approach
this relationship via generalised Artin braid groups. In addition, we shall have recourse to the
“Temperley–Lieb algebra” of type Bn , denoted by TLB n (q, Q) below (it is sometimes referred
a
to as the “blob algebra”), to determine the action of the Xi , since Tn−1
(q) is not naturally a
a
subalgebra of Tn (q), and therefore one does not have restriction. We circumvent this difficulty
by proving that for any Q, there is a pair of natural surjections from Tna (q) to TLB n (q, Q),
and studying restriction from TLB n (q, Q) to TLB n−1 (q, Q). This could be used to study the
na (q), but we do not do this here. In the final Section 11 we
“modular representation theory” of H
interpret our results in terms of a generalisation to the non-generic case of the “multisegments”
of Zelevinsky and Bernstein.
Since the “annular algebras” of V. Jones [16] are quotients of the algebras Tna (q), their
representation theory may be thought of as a subset of the story below. Hence our work throws
light on the connection between the work of Jones on link invariants (cf. [17]) and affine Hecke
algebras.
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2. Some generalised Artin braid groups
Let W be the symmetric group Symn , realised as a Coxeter group generated by the reflections
si in the hyperplanes zi − zi+1 = 0 of V = Cn (i = 1, 2, . . . , n − 1). Write sn for reflection
in the affine hyperplane z1 − zn = 1. Then {s1 , . . . , sn } are Coxeter generators for the affine
Weyl group W a ∼
= W Zn−1 , which may also be thought of as generated by W together
a for the
with translations by vectors (a1 , . . . , an ), with ai ∈ Z and a1 + · · · + an = 0. Write W
n
semidirect product of W with the group Z of all translations by vectors with integer coordinates.
Write v0 = (1, 1, . . . , 1) and denote by p0 the orthogonal projection p0 : V → v0⊥ . Then W and
a , with
W a act irreducibly as Coxeter groups on V0 = v0⊥ , and W a is a normal subgroup of W
quotient Z.
The reflecting hyperplanes of W a acting on V are the hyperplanes zi − zj = k, for
1 i < j n and k ∈ Z. Write M a for the complement of these hyperplanes in V and M0a
for p0 (M a ). Thus, explicitly,
M a = (x1 , x2 , . . . , xn ) ∈ V = Cn | xi − xj ∈
/ Z if i = j .
(2.1) P ROPOSITION (Nguyen [27]). – The fundamental group π1 (M0a /W a ) is isomorphic to
the Artin group associated to the Coxeter system (W a , {s1 , . . . , sn }).
In fact Nguyen proves this for any affine type Coxeter group by giving a “cell decomposition”
of the space M0a /W a . In our case we have explicitly that π1 (M0a /W a ), which we denote
henceforth by ∆n , is generated by elements {σ1 , σ2 , . . . , σn } subject to the relations
(BR)
σi σj = σj σi
if j ≡ i ± 1 (mod n),
σi σi+1 σi = σi+1 σi σi+1
for i = 1, 2, . . . , n,
where the subscripts in (BR) are taken mod n.
(2.2) L EMMA. – The map p0 : M a → M0a is a W a -homotopy equivalence. Hence the quotient
spaces M a /W a and M0a /W a are homotopy equivalent. In particular they have the same
fundamental group.
Proof. – If I is the unit interval and i is the inclusion of M0a in M a , then p0 ◦ i = idM0a , and
0
a
a
a
the map (v, t) → v − (1 − t) v,v
n v0 (M × I → M ) defines a homotopy from i ◦ p0 to idM ,
a
which commutes with the W action for each t ∈ I. ✷
a ⊃ W a be as described above. The map M a /W a → M a /W
a is an
(2.3) L EMMA. – Let W
unramified covering with covering group Z.
a /W a is generated by the element
The proof is easy. Note that the quotient W
τ = cn , (1, 0, 0, . . ., 0) ∈ Symn Zn ,
where cn is the n-cycle (12 . . . n) ∈ Symn . This element has the property that τ n lies in the
a ). It follows from the lemma that M a /W
a may be thought of as the quotient of
centre Z(W
M a /W a by the cyclic group τ .
(2.4) C OROLLARY. – There is an exact sequence
a → Z → 1.
1 → π1 M a /W a → π1 M a /W
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a and its fundamental group. For any topological space Y ,
We now identify the space M a /W
let Xn (Y ) be the space
Xn (Y ) = (y1 , . . . , yn ) ∈ Y n | yi = yj if i = j / Symn .
a ∼
Then evidently M a /W
= Xn (C∗ ). But if MBn is the complex hyperplane
= Xn (C/Z) ∼
n
complement of type Bn (viz. C with the hyperplanes xi ± xj = 0 and xi = 0 removed), and
WBn is the corresponding Weyl group, then clearly
(2.5)
a .
MBn /WBn ∼
= Xn (C∗ ) ∼
= M0a /W
= Xn (C∗ )/{±id} ∼
But Deligne has shown [7] that the space of regular orbits of a finite Coxeter group on its
complexified reflection representation space is its associated generalised Artin braid group (this
is also proved in [27]). Write Γn for the generalised Artin braid group of type Bn . Then Γn has
generators {ξ1 , σ1 , σ2 , . . . , σn−1 }, with relations
σi σj = σj σi
(BRB)
if |i − j| = 1,
σi σi+1 σi = σi+1 σi σi+1
for i = 1, 2, . . . , n − 2,
ξ1 σ1 ξ1 σ1 = σ1 ξ1 σ1 ξ1 ,
ξ1 σi = σi ξ1
if i = 1.
Now paths in Xn (C∗ ) may be regarded as periodic braids, or braids on a thickened cylinder,
as follows. Think of C∗ as the plane with a large hole at 0; choose n points P1 , P2 , . . . , Pn
in C∗ . A path in Xn (C∗ ) may then be regarded as a braid “around the hole”, or on the thickened
cylinder, where each string starts at some Pi and finishes at Pi , where i → i is a permutation of
{1, 2, . . ., n}. If we cut the cylinder open, these braids may be drawn in the plane, and regarded
as “periodic braids”, or cylindrical braids. These may be drawn as depicted in the diagrams in
Fig. 1, in which the two intervals labelled AB are identified by bending the rectangle in towards
the page.
Now let τ ∈ Γn be the “twist” as shown (it corresponds to the projection of a path in
M0a /W a from the base point P to τ P , where τ is the element defined above, such that
a ∼
M a /W
= (M a /W a )/τ ). Further, let σi be the generating braids depicted in the diagram
(i = 1, 2, . . . , n). It is clear that Γn then has a presentation with generators {τ, σ1 , σ2 , . . . , σn },
with relations (BR) for the σi , together with
τ σi τ −1 = σi+1
for i = 1, 2, . . . , n,
where the subscripts are taken mod n.
(2.6) P ROPOSITION. – With the above notation, let
−1
−1
σn−2
. . . σ1−1
ξ1 = τ σn−1
ξi+1 = σi ξi σi
and
for i = 1, 2, . . . , n − 1.
Then the family {ξ1 , σ1 , . . . , σn−1 } generates Γn subject to the relations (BRB), and {ξ1 , . . . , ξn }
generates a free abelian group of rank n.
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Fig. 1.
Proof. – It is clear that {ξ1 , σ1 , . . . , σn−1 } generates Γn , since {τ, σ1 , σ2 , . . . , σn−1 } does, and
τ = ξ1 σ1 . . . σn−1 . We next show that {ξ1 , σ1 , . . . , σn−1 } satisfies the relations (BRB). This is
easy to see from the braid pictures (see Fig. 1), but we shall provide an algebraic proof. For
i = 1, 2, . . . , n, we have
−1
−1
ξi = σi−1 σi−2 . . . σ1 τ σn−1
σn−2
. . . σi−1 .
We shall first show that
σj ξi = ξi σj
(2.6.1)
if j = i − 1, i.
To see (2.6.1), first take i = 1. Then for j > 1,
−1
−1
−1 −1
σj ξ1 = σj τ σn−1
σn−2
. . . σ1−1 = τ σj−1
σn−1 . . . σ1−1
−1
−1
−1 −1
= τ σn−1
. . . σj+1
σj−1 σj−1 σj−1
σj−2 . . . σ1−1
−1
−1 −1 −1
−1
= τ σn−1
. . . σj+1
σj σj−1 σj σj−2
. . . σ1−1
= ξ1 σj .
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J.J. GRAHAM AND G.I. LEHRER
This proves (2.6.1) for i = 1. But since ξi = σi−1 σi−2 . . . σ1 ξ1 σ1 . . . σi−1 , it follows that (2.6.1)
holds for all j > i. Now take j < i − 1. Then
−1
. . . σi−1
σj ξi = σj σi−1 σi−2 . . . σ1 τ σn−1
−1
= σi−1 . . . σj σj+1 σj σj−1 . . . σ1 τ σn−1
. . . σi−1
−1
= σi−1 . . . σj+1 σj σj+1 σj−1 . . . σ1 τ σn−1
. . . σi−1
by (BR)
= ξi σj .
This proves (2.6.1). We next show that
ξ1 σ1 ξ1 σ1 = σ1 ξ1 σ1 ξ1 .
(2.6.2)
Let
−1
−1
−1
βn = σn−1 σn−2
. . . σ1−1 σn−1
σn−2
. . . σ1−1 .
We shall show by induction on n that
−1
−1
−1
−1
βn = σn−2
σn−1
σn−3
σn−2
. . . σ1−1 σ2−1 .
(2.6.2.1)
If n = 2, both sides of (2.6.2.1) are equal to 1. In general,
−1
−1
−1
−1
−1
βn = σn−1 σn−2
σn−1
σn−3
. . . σ1−1 σn−2
σn−3
. . . σ1−1
−1
−1
−1
−1
−1
= σn−2
σn−1
σn−2 σn−3
. . . σ1−1 σn−1
σn−2
. . . σ1−1
−1
−1
= σn−2
σn−1
βn−1 ,
which proves (2.6.2.1).
−1
−1
−1
Next, observe that if γn = σn−2
. . . σ1−1 σn−1
σn−2
. . . σ2−1 , then γn = βn . For n = 3,
−1 −1
−1
−1
γn = βn = σ1 σ2 . In general, γn = σn−2 σn−1 γn−1 , and so
γn = βn
(2.6.2.2)
for all n, by induction on n.
To prove (2.6.2), we now have
−1
−1
−1
−1
σ1 ξ1 σ1 ξ1 = σ1 τ σn−1
σn−2
. . . σ2−1 τ σn−1
σn−2
. . . σ1−1
−1
−1
−1
−1
−1
= σ1 τ 2 σn−2
σn−3
. . . σ1−1 σn−1
σn−2
σn−3
. . . σ1−1
−1
−1
−1
−1
= τ 2 σn−1 σn−2
σn−3
. . . σ1−1 σn−1
σn−2
. . . σ1−1
= τ 2 βn .
But
−1
−1
−1
−1
ξ1 σ1 ξ1 σ1 = τ σn−1
σn−2
. . . σ2−1 τ σn−1
σn−2
. . . σ2−1
−1
−1
−1
−1
= τ 2 σn−2
σn−3
. . . σ1−1 σn−1
σn−2
. . . σ2−1
= τ 2 γn
= τ 2 βn
by (2.6.2.2)
= σ1 ξ1 σ1 ξ1 ,
which proves (2.6.2).
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It follows immediately from (2.6.2) and (2.6.1) that the family {ξ1 , σ1 , . . . , σn−1 } satisfies
the relations (BRB). But the relations (BR) and τ σi τ −1 = σi+1 are similarly shown to follow
from (BRB), whence we have a presentation of Γn .
It remains to show that the ξi commute with each other. Note that (2.6.2) says that ξ1 ξ2 = ξ2 ξ1 .
In general, suppose that we know ξi ξj = ξj ξi for all j > i, 1 i < k. Since ξk = σk−1 ξk−1 σk−1 ,
if j > k, ξj commutes with σk−1 by (2.6.1), and with ξk−1 by induction, and hence with ξk .
Hence by induction the ξi all commute. To see that there are no relations among the ξi , one may
use the braid picture as follows. In any cylindrical braid of the form ξ1m1 . . . ξnmn , note that each
string joins points on the top and bottom of the cylinder which have the same label, since this
is true of each ξi . Given the braid, the indices mi may be recovered as the number of times the
relevant string winds around the cylinder. ✷
(2.7) C OROLLARY. – The exact sequence (2.4) is realised as
1 → ∆n → Γn → Z → 1,
where ∆n → Γn is inclusion and Γn → Z is the map taking τ r σin11 . . . σinll ∈ Γn to r ∈ Z.
Our final result in this section is
(2.8) L EMMA. – We have the following relation in Γn .
τ n = ξ1 ξ2 . . . ξn .
−1
−1
Proof. – Since ξ1 = τ σn−1
σn−2
. . . σ1−1 and ξi+1 = σi ξi σi , one shows easily by induction on i
that for 1 i n − 1,
−1
−1
−1
ξ1 ξ2 . . . ξi = τ i σn−i
. . . σ1−1 σn−i+1
. . . σ2−1 . . . σn−1
. . . σi−1 .
−1
Hence ξ1 ξ2 . . . ξn−1 = τ n−1 σ1−1 . . . σn−1
= τ n ξn−1 .
✷
3. Affine Hecke algebras of type A
Let R be a commutative ring, denote by R× the group of invertible elements of R, and let
q ∈ R× . We maintain the notation of the last section, so that Γn is the Artin braid group B(Bn )
of type Bn and ∆n is the Artin braid group B(A
n−1 ) of type An−1 , regarded as the subgroup
of Γn generated by σ1 , . . . , σn . Denote by RΓn the group ring of Γn over R.
(3.1) D EFINITION. – Let Si be the element Si = (σi − q)(σi + q −1 ) of RΓn (i = 1, 2, . . . , n).
na (q) of GLn over R is defined by
The affine Hecke algebra H
a (q) = RΓn /S1 .
H
n
Note that since S1 , . . . , Sn are all conjugate in RΓn , the ideal S1 is equal to S1 , . . . , Sn .
a (q) be the natural map. We then write
Let η : RΓn → H
n
(3.2)
η(σi ) = Ti
for i = 1, . . . , n,
η(ξi ) = Xi
for i = 1, . . . , n,
η(τ ) = V.
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a (q), many of which may
The next proposition collects some well known facts concerning H
n
be found in §3 of [24].
(3.3) P ROPOSITION. –
a (q), which has R-basis
(i) The elements T1 , . . . , Tn generate a subalgebra Hna (q) of H
n
a∼
n−1
{Tw | w ∈ W = Symn Z
}, where, if w = si1 . . . si is a reduced expression for w ∈ W a ,
Tw = Ti1 . . . Ti . We refer to this as the “unextended” Hecke algebra of type A
n−1 .
a
(ii) The elements T1 , . . . , Tn−1 generate a subalgebra Hn (q) of Hn (q) which has (finite)
R-basis {Tw | w ∈ W ∼
= Symn }.
na (q) ∼
(iii) We have H
= RZ ⊗ Hna (q) ∼
= RV ⊗ Hna (q), where the tensor product is twisted,
a
−1
using the action of V on Hn (q): V Ti V
= Ti+1 , where the subscript is taken mod n.
±1
a
∼
(iv) We have Hn (q) = Hn (q) ⊗ R[X1 , . . . , Xn±1 ] as R-module, and the multiplication is
given by the “Bernstein relations”: for i ∈ {1, . . . , n − 1}, write si for the corresponding simple
reflection in W and si f for the image of f ∈ R[X1±1 , . . . , Xn±1 ] under the natural action of
W∼
= Symn . Then
Ti f −
f − (si f )
si f Ti = q − q −1
−1 .
1 − Xi Xi+1
a (q)) is the ring of symmetric functions in the X ±1 . Equivalently,
(v) The centre Z(H
n
i
±1
na (q)) = R[Σ , . . . , Σ±1
Z(H
n ], where Σi is the ith elementary symmetric function in the Xi .
1
Note that some authors use notation which results in the denominator of (iv) above being
1 − Xi+1 Xi−1 .
We remark that to prove the relation in (iv) from those given in Section 2 for the braid group,
one observes that the relation is linear in f , and hence need only be proved for monomials;
moreover one easily shows that if the relation holds for f1 and f2 , then it holds for f1 f2 . Thus
one is reduced to proving the relation for f = Xj , which is easy.
In addition to the algebras above, we shall need to consider the (finite rank) Hecke algebra
of type B, which arises as follows. Let WB := Symn (Z/2Z)n be the hyperoctahedral group.
This is generated as Coxeter group by {s1 , . . . , sn−1 }, together with another generator s0 . The
generators {s0 , s1 , . . . , sn−1 } are involutions, and satisfy the relations analogous to (BRB) above.
Let Q ∈ R× . The Hecke algebra HB n (q, Q) of type Bn with parameters (q, Q) is defined as
a (q)/ (X1 − Q) X1 + Q−1
HB n (q, Q) = H
n
= RΓn / (ξ1 − Q) ξ1 + Q−1 , (σ1 − q) σ1 + q −1 .
(3.4) P ROPOSITION. – Let
a (q) → HB n (q, Q)
ηQ : H
n
a (q) under ηQ
be the natural map. Write Ti ∈ HB n (q, Q) for the image of Ti ∈ H
n
(i = 1, . . . , n − 1) (relying on the context to distinguish between them), and write T0 = ηQ (X1 ).
Then HB n (q, Q) has R-basis {Tw | w ∈ WB}, where, if w = si1 . . . si is a reduced expression
for w ∈ WB , Tw = Ti1 . . . Ti .
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The relationship among the various algebras introduced so far is illustrated in the commutative
diagram below.
Hn (q)
incl
R∆n
(3.5)
η
Hna (q)
incl
RΓn
incl
η
a (q)
H
n
ηQ
HB n (q, Q)
The relations discussed in Section 2 for the braid groups Γn and ∆n may be interpreted in the
Hecke algebras as follows.
na (q).
(3.6) L EMMA. – The following relations hold in H
(3.6.1)
(3.6.2)
V n = X1 X2 . . . Xn ,
V = X1 T1 T2 . . . Tn−1 .
If we write V for ηQ (V ) and adopt the notation of (3.4), we also have (in HB n (q, Q))
(3.6.3)
V = T0 T1 . . . Tn−1 .
4. Affine and finite dimensional Temperley–Lieb algebras
Let W = s1 , . . . , sn−1 ∼
= Symn as above and write
Wi = si , si+1 ∼
= Sym3
for i = 1, 2, . . . , n − 2.
a (q) by
Define the element Ei ∈ Hn (q) ⊂ Hna (q) ⊂ H
n
Ei =
q (w) Tw
w∈Wi
ˆ denote the ideal of H a (q) (resp.
where 4(w) denotes the usual length function. Let I (resp. I)
n
a
Hn (q)) generated by E1 . Note that since the Ei are all conjugate, this is the same as the ideal
generated by all the Ei .
a (q) are defined
(4.1) D EFINITION. – The affine Temperley–Lieb algebras TLan (q) and TL
n
by
TLan (q) = Hna (q)/I,
a (q) = H
a (q)/I.
ˆ
TL
n
n
It is known (cf. [14,15]) that if Ci = −(Ti + q −1 ) ∈ Hna (q) (i = 1, . . . , n), then in Hna (q),
Ci Ci+1 Ci − Ci = Ci+1 Ci Ci+1 − Ci+1 = −q 3 Ei , where the indices are taken mod n. If we
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abuse notation by writing Ci ∈ TLan (q) for the image of Ci ∈ Hna (q) under the natural map, it
follows easily that TLan (q) is generated by {C1 , . . . , Cn } subject to the relations
Ci2 = δq Ci ,
(TL)
Ci Ci±1 Ci = Ci ,
Ci Cj = Cj Ci
if |i − j| 2 and {i, j} = {1, n},
where, for any element x ∈ R× , δx := −(x + x−1 ).
Moreover it is easy to see that (cf. (3.3)(iii) above, or [15, §2])
(4.2)
a (q) ∼
TL
= RV ⊗ TLan (q),
n
where V permutes the Ci cyclically.
a (q), we shall require the algebra T a (q) which
Now in addition to the algebras TLan (q) and TL
n
n
was defined in [14, (2.7)] and referred to there (loc. cit.) as “the affine Temperley–Lieb algebra”.
This is defined as an algebra of diagrams or, more accurately, as the algebra of morphisms:
n → n in the category Ta (see [14, (2.5)]) and TLan (q) is identified [14, (2.9)] as the subalgebra
of Tna (q) spanned by the “non-monic diagrams : n → n of even rank”, together with the identity.
It also occurs independently in the work of Green [11] and Fan–Green [8]. We shall need to make
some use of the diagrammatic description in this work; details may be found in [14], but a good
approximation to the picture is obtained if one thinks of affine diagrams as arcs drawn on the
surface of a cylinder joining 2n marked points, n on each circle component of the boundary, in
pairs. The arcs must not intersect, and diagrams are multiplied by concatenation in the usual way.
These diagrams are represented by periodic diagrams drawn between two horizontal lines, each
diagram being determined by the “fundamental rectangle”, from which the cylinder is obtained
by identifying vertical edges. In this interpretation, the generators {f1 , . . . , fn , τ } of Tna (q) are
represented by the diagrams in Fig. 2. The elements {f1 , . . . , fn } of the algebra Tna (q) satisfy the
relations (TL), with Ci replaced by fi , and it is noted in [14, (2.9)] that these generate an algebra
isomorphic to TLan (q). Further, τ fi τ −1 = fi+1 , where the index is taken mod n.
Fig. 2.
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(4.3) P ROPOSITION. –
a (q) → T a (q) (α ∈ R× ), defined by φα (Ci ) = fi
(i) There is a family of surjections φα : TL
n
n
and φα (V ) = ατ . Each φα restricts to a monomorphism on TLan (q).
a (q), where
(ii) The kernel of φα is generated by the element να ∈ TL
n
να = α−2 V 2 Cn−1 − C1 C2 . . . Cn−1 = α−2 C1 V 2 − C1 C2 . . . Cn−1 .
(iii) If R is an algebraically closed field of characteristic prime to n, any irreducible finite
a (q) is the pullback via φα (for some α ∈ R× ) of an irreducible
dimensional representation of TL
n
a
representation of Tn (q).
Proof. – The first part of (i) follows immediately from the relations above, while the second
follows from the fact (cf. [14, 2.9]) that (TL) gives a presentation of TLan (q). Next, one verifies
easily that τ 2 fn−1 = f1 f2 . . . fn−1 in Tna (q) (see [15, 1.11]), which shows that να ∈ Ker φα . The
fact that να generates the kernel may be found in [11] or [8]. This relation also appears in [16].
The statement (iii) may be proved using the argument of Theorem 2.6 in [15]. ✷
In [14], we defined cell modules Wt,z (n) for the algebra Tna (q), (where t ∈ Z, 0 t n,
t + n ∈ 2Z, and z ∈ R× ) and when R is an algebraically closed field of characteristic zero,
completely determined their composition factors and multiplicities. Our purpose here is to
interpret these results for the pullbacks φ∗α Wt,z (n). To identify these pullbacks as standard
a (q) up to Grothendieck equivalence (cf. [19] or [31]), we need to determine the
modules for H
n
action of the translation elements Xi on the modules, and for this we shall require the Temperley–
Lieb algebra TLB n (q, Q) of type Bn .
5. The Temperley–Lieb algebras of type B
This algebra has been studied by mathematical physicists [25,26], where it is referred to as
the “blob algebra”, and in [29] (see also [6], and the references there). We shall present here
the main facts which we require concerning TLB n (q, Q), relying for general background on
op. cit. Our notation continues from Section 3 above, and we start with an algebraic definition
of TLB n (q, Q).
Note first that the Hecke algebra HB n (q, Q) has an R-algebra homomorphism
ε : HB n (q, Q) → R,
defined on the generators Ti (i = 0, 1, . . . , n − 1, see (3.4)) by
ε(T0 ) = Q,
ε(Ti ) = q
for 1 i n − 1.
As above, let s0 , s1 , . . . , sn−1 be the simple generators of the hyperoctahedral
group WB , and
write Wi = si , si+1 for i = 0, 1, . . . , n − 2. Then define Ei = w∈Wi ε(Tw )Tw . For i = 0,
these Ei coincide with the Ei of Section 4, and they are all conjugate in HB n (q, Q). For i > 0,
let Ci = −(Ti + q −1 ), and let C0 = −(T0 + Q−1 ). The next lemma summarises several relevant
relations in HB n (q, Q).
(5.1) L EMMA. –
(i) For i = 1, . . . , n − 2, Ci Ci+1 Ci − Ci = −q 3 Ei .
(ii) For i = 2, . . . , n − 1, Ci Ci−1 Ci − Ci = −q 3 Ei .
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q
−2 −2
(iii) Let X = C1 C0 C1 − κC1 , where κ = Q
+Q
Q E0 .
q . Then XC0 = C0 X = q
2
2
(iv) We have Ci = δq Ci for i = 0, while C0 = δQ C0 .
The proofs are simple computations in HB n (q, Q).
(5.2) D EFINITION. – The Temperley–Lieb algebra TLB n (q, Q) of type Bn with parameters
(q, Q) is defined as
TLB n (q, Q) := HB n (q, Q)/E1 , X = HB n (q, Q)/E1 , E2 , . . . , En−2 , X.
na (q) → TLB n (q, Q) contains the kernel of the
Note that the kernel of the natural map H
a
a
natural map Hn (q) → TLn (q). Hence the former map factors through a surjection
a (q) → TLB n (q, Q).
TL
n
This is reflected in the diagram (5.6) below.
Denote by lower case letters the images in TLB n (q, Q) of the corresponding elements of
a (q) or HB n (q, Q)). Thus in particular we write ti , ci respectively for the images
a
Hn (q) (or TL
n
in TLB n (q, Q) of Ti , Ci ∈ HB n (q, Q) under the natural map (i = 0, 1, 2, . . ., n). Similarly, we
have elements xi (i = 1, . . . , n) and v ∈ TLB n (q, Q). The next statement is easy to verify from
the foregoing discussion.
(5.3) P ROPOSITION. –
(a) The Temperley–Lieb algebra TLB n (q, Q) is generated as R-algebra by the family
{c0 , c1 , . . . , cn−1 } subject to the relations
c20 = δQ c0 ,
c2i = δq ci
(TLB)
for 1 i n − 1,
ci ci+1 ci = ci
for 1 i n − 2,
ci ci−1 ci = ci
for 2 i n − 1,
c1 c0 c1 = κc1 ,
q
−1
where κ = Q
+Q
) for x ∈ R× .
q and δx = −(x + x
(b) The elements c1 , . . . , cn−1 generate a subalgebra of TLB n (q, Q) which is isomorphic to
the usual Temperley–Lieb algebra TLn (q) of type A
n−1 .
(c) The following relations hold in TLB n (q, Q).
(i) t0 = −(c0 + Q−1 ) and ti = −(ci + q −1 ) for i = 1, . . . , n − 1.
(ii) If v ∈ TLB n (q, Q) is the image of V ∈ HB n (q, Q) under the natural map, then
v = t0 t1 . . . tn−1 .
(iii) x1 x2 . . . xn = v n .
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A diagram of maps relating the algebras we have now introduced is as follows.
Hn (q)
η1
TLn (q)
incl
R∆n
(5.6)
η
incl
RΓn
Hna (q)
incl
η2
TLan (q)
incl
η
a (q)
H
n
incl
η3
a (q)
TL
n
ηQ
HB n (q, Q)
φα
Tna (q)
γQ
η4
TLB n (q, Q)
where the ηi are the natural surjections and γQ exists because the kernel of η4 ◦ ηQ contains
a (q).
a (q), which generates the kernel of η3 . Thus η4 ◦ ηQ factors through TL
E1 ∈ H
n
n
The algebra TLB n (q, Q) has a description in terms of marked diagrams (see [25,26,29])
which we now describe, because this description is convenient for the discussion of cell modules.
We shall use the language of [14] for diagrams. Recall that if t, n are positive integers of the same
parity, a finite (planar) diagram µ : t → n is represented by a set of non-intersecting arcs which
are contained in the “fundamental rectangle” (see below). These arcs divide the fundamental
rectangle into regions, among which there is a unique “left region” as shown below.
A marked diagram is a (finite planar) diagram, where the interior of the boundary arcs of the
leftmost region may be marked with one or more • symbols (“marks”) (see below).
The R-linear combinations of diagrams from t to n constitute the morphisms in the
Temperley–Lieb category T, where the objects are the non negative integers Z0 . Composition
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is defined by concatenation of diagrams, with closed loops being deleted and replaced by the
scalar δq . In particular, HomT (n, n) ∼
= TLn (q). Marked diagrams may be similarly concatenated
according to rules we shall now state; this produces a new category, TB, the Temperley–Lieb
category of type B. The composition rules are as follows.
A marked diagram is proper if it has no loops and each arc has at most one mark. The
following rules reduce the concatenation of any two diagrams to an R linear combination of
proper diagrams:
(5.7.1)
(5.7.2)
(5.7.3.)
If µ is a diagram and L is a loop with no marks, µ L = δq µ.
Q
q
+ .
Q
q
If some arc of µ has more than one mark and µ is the diagram obtained
If, in (i), L has one mark, µ L = κµ, where κ =
by removing a mark from the arc concerned, then µ = δQ µ .
Now consider the following marked diagrams from n to n.
(5.8) P ROPOSITION. – The diagrams labelled c0 , c1 , . . . , cn−1 satisfy the relations for the
generators of TLB n (q, Q) given in (5.3). Moreover TLB n (q, Q) is faithfully represented in this
way as EndTB (n) in the category TB.
Proof. – The relations among diagrams ci are easily checked using the rules (5.7). It follows
that there is a homomorphism : TLB n (q, Q) → EndTB (n), which is easily seen to be surjective,
since any proper marked diagram n → n can be obtained by composing the ci . The injectivity of
this homomorphism follows from arguments in [29] or [12]. ✷
Let us now focus attention on the bottom right part of the commutative diagram (5.6). Recall
a (q) is mapped to T0 ∈ TLB n (q, Q). But for any element
that under the map ηQ , X1 ∈ H
n
×
µ ∈ R , we may deform both ηQ and γQ to ηQ,µ and γQ,µ respectively, where ηQ,µ (X1 ) = µT0 ,
ηQ,µ (Ti ) = Ti for 1 i n − 1, γQ,µ (V ) = µγQ (V ) = µt0 t1 . . . tn−1 and γQ,µ (Ci ) = ci for
1 i n − 1. We then obtain a “deformed” version of the bottom right part of the commutative
diagram (5.6) as follows.
na (q)
H
(5.9)
η3
ηQ,µ
HB n (q, Q)
a (q)
TL
n
φα
Tna (q)
γQ,µ
η4
TLB n (q, Q)
na (q) of cell modules of Tna (q) by realising the latter as
We shall study the pullbacks to H
pullbacks of cell modules of TLB n (q, Q) via a homomorphism yet to be defined. With the
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objective of defining a map : Tna (q) → TLB n (q, Q) which makes the diagram (5.9) commute,
we shall prove
(5.10) T HEOREM. – Let ti , ci and v be the elements of TLB n (q, Q) defined just before (5.3)
above. Then for n 2 we have
c1 v 2 = −q −(n−2) c1 c2 . . . cn−1 .
(5.10.1)
Proof. – This will be by induction on n. We first verify the statement for n = 2. The left side
of (5.10.1) is then
c1 (t0 t1 )2
= c 1 t0 t1 t0 t1
= c 1 t1 t0 t1 t0
since the ti satisfy the braid relations (BRB)
= qc1 t0 t1 t0 since ci ti = qci
= −qc1 c0 + Q−1 c1 + q −1 c0 + Q−1
= −qc1 c0 c1 c0 + q −1 δQ c0 + Q−1 (c0 c1 + c1 c0 ) + 2q −1 Q−1 c0 + Q−2 c1 + q −1 Q−2
= −q κc1 c0 + q −1 δQ c1 c0 + Q−1 (κc1 + δq c1 c0 )
+ 2q −1 Q−1 c1 c0 + Q−2 δq c1 + q −1 Q−2 c1
= −qc1 c0 κ + q −1 δQ + Q−1 δq + 2q −1 Q−1 − qc1 Q−1 κ + Q−2 δq + q −1 Q−2
= −c1 ,
which proves (5.10.1) for the case n = 2.
Now suppose n 3. Then
c1 v 2 = c1 t0 t1 . . . tn−1 t0 t1 . . . tn−1
= c1 t0 t1 . . . tn−2 t0 t1 . . . tn−3 tn−1 tn−2 tn−1
= c1 t0 t1 . . . tn−2 t0 t1 . . . tn−3 tn−2 tn−1 tn−2
by the braid relations (BRB)
2
= c1 (t0 t1 . . . tn−2 ) tn−1 tn−2
= −q −(n−3) c1 c2 . . . cn−2 tn−1 tn−2 by induction on n
= −q −(n−3) c1 c2 . . . cn−2 cn−1 + q −1 cn−2 + q −1
= −q −(n−3) c1 c2 . . . cn−3 cn−2 + q −1 cn−2 cn−1 + q −1 δq cn−2 + q −2 cn−2
= −q −(n−2) c1 c2 . . . cn−1
by (5.3),
which completes the proof. ✷
(5.11) C OROLLARY. – For each element β ∈ R such that β 2 = −q n−2 , there is a (unique)
surjective homomorphism gβ : Tna (q) → TLB n (q, Q) such that gβ (fi ) = ci for i = 1, 2, . . . , n − 1,
and gβ (τ ) = βv. If α, µ satisfy α−1 µ = β, then the following diagram commutes.
na (q)
H
(5.11.1)
η3
ηQ,µ
HB n (q, Q)
γQ,µ
η4
φα
a (q)
TL
n
gβ
TLB n (q, Q)
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a (q) to ci ∈ TLB n (q, Q),
Proof. – For i = 1, 2, . . . , n − 1, the surjection γQ,µ takes Ci ∈ TL
n
while γQ,µ (V ) = µv = µt0 t1 . . . tn−1 . Hence by Theorem (5.10),
C1 V 2 + µ2 q −(n−2) C1 C2 . . . Cn−1 ∈ Ker(γQ,µ ).
But Ker(φα ) is generated by
να = α−2 C1 V 2 − C1 C2 . . . Cn−1 .
Hence Ker(γQ,µ ) ⊃ Ker(φα ), provided that
α2 = −µ2 q −(n−2) .
(5.11.2)
a (q) → TLB n (q, Q) factors through
If this condition is satisfied, γQ,µ : TL
n
a (q) → T a (q);
φα : TL
n
n
i.e., there is a map gβ : Tna (q) → TLB n (q, Q) which makes the diagram (5.11.1) commute.
For this map, we have by commutativity, gβ (fi ) = γQ,µ (Ci ) = ci for i = 1, 2, . . . , n − 1, and
gβ (τ ) = γQ,µ (α−1 V ) = βv, where β = α−1 µ satisfies (5.11.2), i.e. β 2 = −q n−2 . ✷
Note that the homomorphism gβ depends of course on Q. We shall write gβ = gβ,Q when we
need to refer to this dependence.
Suppose now that we have a triple (α, µ, β) of elements of R× such that β 2 = −q n−2 and
β = α−1 µ. By (5.11), we have a commutative diagram
ψα
na (q)
H
ξQ,µ
(5.12)
Tna (q)
gβ
TLB n (q, Q)
where, in the notation of (5.11.1), ψα = φα ◦ η3 and ξQ,µ = γQ,µ ◦ η3 = η4 ◦ ηQ,µ .
For future reference, we summarise the definitions of the maps in (5.12) in the following
equations.
ξQ,µ (X1 ) = µt0 ,
ξQ,µ (Ti ) = ti
(5.13)
for i = 1, 2, . . . , n − 1,
ψα (V = X1 T1 . . . Tn−1 ) = ατ,
ψα (Ci = −(Ti + q −1 )) = fi
for i = 1, 2, . . . , n − 1,
gβ (fi ) = ci = −(ti + q −1 )
for i = 1, 2, . . . , n − 1,
gβ (τ ) = βv = βt0 t1 . . . tn−1 .
Our next objective is to identify the pullbacks via ψα of the cell modules Wt,z (n) of
a (q) up to Grothendieck equivalence. We shall do this
as standard modules Ms,N of H
n
Tna (q)
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by identifying Wt,z (n) as the pullback of a cell module Wt (n) of TLB n (q, Q) via gβ for
appropriate Q, and using the commutative diagram (5.12). We therefore need to discuss the
cellular structure of TLB n (q, Q), which we do in the next section.
6. Cell modules for Tna (q) and TLB n (q, Q)
The cell modules Wt,z (n) of Tna (q) were defined and analysed in [14]. We shall freely use the
notation and language of that work here. There is an entirely analogous “cellular theory” for the
algebras TLB n (q, Q) which we shall now sketch. Recall [13] that to specify a cellular structure
for TLB n (q, Q), we require (i) a poset T , (ii) for each t ∈ T , a set M (t), and (iii) an injection
C
t
} of TLB n (q, Q) which
t∈T M (t) × M (t) → TLB n (q, Q), whose image is an R-basis {CS,T
satisfies
t
=
ra (S , S)CSt ,T + lower terms (corresponding to t < t)
aCS,T
S ∈M(t)
(a ∈ TLB n (q, Q), ra (S , S) ∈ R).
We now specify the various elements of a cellular structure for TLB n (q, Q). Take
T = t ∈ Z | |t| n, t ≡ n (mod 2) ,
partially ordered as follows: t s if |t| < |s| or |t| = |s| and t s.
To define M (t), first take t ∈ T , t 0. Then M (t) is the set of monic diagrams D : t → n
with no marked through strings, where “monic” means that there are t through strings, as in [14],
where it is shown that this is equivalent to D being a monic morphism in the category-theoretic
sense. In general, let M (t) = M (|t|). Then C : M (t) × M (t) → TLB n (q, Q) is defined as
t
follows. Let S, T ∈ M (t). For t 0, define CS,T
= S ◦ T ∗ , where ∗ denotes reflection in a
t
horizontal axis. For t < 0, define CS,T = S ◦ co ◦ T ∗ , where c0 = c0 (t) : t → t is the generator
shown above (5.8). This is the diagram S ◦ T ∗ , with the leftmost through string marked. The
cellular axioms above are easily checked.
The cell modules Wt (n) are now defined in complete analogy with the Wt,z (n) of [14]. For
any t ∈ T , Wt (n) has basis M (t). If t 0, TLB n (q, Q) acts via composition in the category
TB; explicitly, if D ∈ M (t) and ω ∈ TLB n (q, Q), then ω.D = ωD (composition in TB) if
ω ◦ D ∈ M (t), and ω.D = 0 otherwise.
For t < 0, one may think of Wt (n) as having basis the set {D ◦ c0 (t) | D ∈ M (|t|)} of monic
diagrams : t → n in TB with the leftmost through string marked. Then the action of TLB n (q, Q)
is essentially multiplication in TB, as in the case t 0. Thus if t < 0, then ω.(D ◦ c0 (t)) = 0 if
ω ◦ D is not monic, while ω.(D ◦ c0 ) = ωDc0 (composition in TB) if ω ◦ D ismonic.
n
, which is
It is easily seen (cf. [29]) that the dimension (i.e. rank over R) of Wt (n) is n−|t|
2
the same as that of Wt,z (n). We now wish to identify the inflation of Wt (n) via the surjection
gβ : Tna(q) → TLB n (q, Q).
(6.1) T HEOREM. – Suppose R is any commutative ring and suppose that q, Q are elements
of R× , and that δq = −(q + q −1 ) is not a zero-divisor. Let β ∈ R× satisfy β 2 = −q n−2 and
let gβ,Q = gβ : Tna (q) → TLB n (q, Q) be the surjection defined in (5.11). For t ∈ Z such that
1
0 t n and t ≡ n (mod 2), write zt = (−1)t βQ−1 q − 2 (n+t−2) . If Wt (n) is the cell module
∗
for TLB n (q, Q) described above, then the inflation gβ Wt (n) (i.e. the pullback via gβ of Wt (n))
is isomorphic to Wt,z (n), where z = zt .
The proof will depend on a sequence of lemmas.
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(6.2) Remark. – Observe first that the finite rank Temperley–Lieb algebra TLn (q) is naturally
a subalgebra of both Tna (q) and TLB n (q, Q); in Tna (q) it is generated by f1 , . . . , fn−1 , and
has R-basis consisting of “finite diagrams” from n to n, i.e., diagrams with no arcs crossing
the boundary of the fundamental rectangle (see [14, §2]). In TLB n (q, Q), it is generated by
c1 , . . . , cn−1 and has basis consisting of unmarked diagrams from n to n. These two sets of
diagrams may be identified in the obvious way, and relations (5.13) show that gβ maps a diagram
in TLn (q) ⊂ Tna (q) to the same diagram, regarded as a diagram in TLB n (q, Q). A similar
observation applies more generally to affine and marked diagrams t to n, where finite and
unmarked diagrams respectively may be identified with each other.
Recall [14] that an affine diagram from t to n is monic if it has t through strings. Let Xt
be the Tna (q)-module with basis all monic affine diagrams : t → n, with Tna (q) action given by
composition in the category Ta , modulo diagrams with fewer than t through strings. Thus Xt
may be thought of as a quotient of the left Tna (q)-module HomTa (t, n) by the submodule spanned
by diagrams with fewer than t through strings. The Tna (q) module Wt,z (n) is defined as the
quotient of Xt by the ideal
(6.3)
Iχ := {γτt − χγ}
for γ ∈ Xt , where χ = χz = z if t = 0 and χ = χz = z + z −1 if t = 0.
We shall define a homomorphism θ : Xt → gβ∗ Wt (n) which we shall prove factors through
Wt,z (n) for appropriate z. To do this, we require the following diagrams, which, being finite,
may (and will) be alternately thought of as lying in the categories Ta and TB.
(6.4) D EFINITION. – First suppose t > 0. Define monic finite diagrams Dt , Dt : t → n as
depicted below.
In Dt , (t > 0) the t through strings are joined to the rightmost t top nodes, while in Dt they
are joined to the rightmost t − 1 and leftmost top nodes. Note that Dt and Dt are finite diagrams
and that
(6.4.1)
Dt∗ Dt = (Dt )∗ Dt = idt .
When t = 0 the corresponding diagrams are as below.
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In this case
(D0 )∗ D0 = D0∗ D0 = δq id0 ,
(6.4.2)
(6.5) D EFINITION. – Let Xt be the Tna (q)-module with basis the set of all monic diagrams
t → n (as explained above). Define θ : Xt → Zt := gβ∗ Wt (n) by the formula
θ(γ) = gβ (γDt∗ )Dt ,
(6.5.1)
where γ ∈ Xt , and Dt∗ and Dt are regarded as diagrams in Ta and TB respectively (see
Remark (6.2)).
(6.6) L EMMA. –
(i) (cf. Remark (6.2)) If γ ∈ Xt is a finite diagram, then provided that t > 0, we have θ(γ) = γ;
i.e., θ(γ) is the same diagram as γ, regarded as a marked diagram in Wt (n). If t = 0, then
θ(γ) = δq γ.
(ii) Suppose that for some element χ ∈ R, we know that θ(γτt ) = χθ(γ) for all finite diagrams
γ ∈ Xt . Then
θ(γτt ) = χθ(γ)
for all γ ∈ Xt ; i.e., Ker(θ) ⊃ Iχ .
Proof. – Suppose γ ∈ Xt is a finite diagram. Then γDt∗ is also finite, and hence is in
TLn (q) ⊂ Tna (q). By Remark (6.2), any diagram in TLn (q) is mapped by gβ to the same
diagram, regarded as an element of TLB n (q, Q), whence θ(γ) = γDt∗ Dt = γ, which proves (i)
for t > 0. The same argument may be used for the case t = 0, taking into account that
D0∗ D0 = δq id0 .
(ii) Clearly any diagram γ ∈ Xt has a factorisation γ = ωγ0 , where γ0 ∈ Xt is finite, and ω is
a diagram in Tna (q) (in fact more is true: Xt is cyclic as Tna (q)-module, generated by any one of
many finite diagrams). Hence
θ(γτt ) = θ(ωγ0 τt )
= gβ (ω)θ(γ0 τt ) since θ is a Tna (q)-module homomorphism
= χgβ (ω)θ(γ0 )
= χθ(ωγ0 )
= χθ(γ),
as required. ✷
(6.7) C OROLLARY. – The homomorphism
θ : Xt → gβ∗ Wt (n) = Zt
of Tna (q) modules is surjective if t > 0, and has image precisely δq Zt when t = 0.
Proof. – By (6.6), if t > 0, the image of θ contains all finite diagrams t → n. But these generate
Wt,z (n) as Tna (q)-module, whence the first statement. If t = 0, then again by (6.6), for any finite
diagram γ, θ(γ) ∈ δq Zt . Since the finite diagrams generate Zt as Tna (q)-module, it follows that
the image is contained in δq Zt . But since the image contains δq γ for each finite diagram, the
result is clear. ✷
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We shall show that θ factors through Wt,z (n) for appropriate z ∈ R× using the method of
(6.6)(ii), i.e. by showing that Ker(θ) ⊃ Iχ for χ = χz , z = zt . We therefore investigate the right
action of τt on Xt .
Recall (5.13) that gβ (τn ) = βt0 t1 . . . tn−1 , where τn is thought of as an element of Tna (q).
Thus to carry through the strategy of (6.6), we shall need the following computation in TB. We
adopt the following notation for diagrams in TB: we say that a diagram is positive if the leftmost
through string is unmarked, and negative otherwise. If E is a positive diagram (e.g. in Wt (n)),
Ė will denote the diagram obtained from E by marking the leftmost through string.
(6.8) L EMMA. – Suppose t > 0. Then in the category TB, we have
βtn−1 tn−2 . . . t1 t0 Dt = zt (Dt + QḊt + E)
where Dt , Dt are the diagrams defined above and E is a linear combination of diagrams with < t
through strings. The right side of this equation is to be thought of as an element of HomTB (t, n).
Proof. – Recall that for 1 i n − 1, ti = −(ci + q −1 ), while t0 = −(c0 + Q−1 ). Hence
t0 Dt = −Ḋt − Q−1 Dt . Thus we need to evaluate tn−1 tn−2 . . . t1 Dt and tn−1 tn−2 . . . t1 Ḋt .
For the former, we have t1 Dt = −Dt (n − 2) − q −1 Dt , where Dt (n − 2) is the diagram
obtained from Dt by moving the leftmost through string past the leftmost horizontal arc. Now
c2 Dt (n − 2) = Dt and c2 Dt = δq Dt , from which it follows that
t2 t1 Dt = q −1 Dt (n − 2).
(6.8.1)
Repeating this argument k times, where 2k = n − t, we obtain
t2k t2k−1 . . . t2 t1 Dt = q −k Dt .
(6.8.2)
But for i > 2k one sees easily that ti Dt ≡ −q −1 Dt modulo diagrams with fewer than t through
strings. Hence
tn−1 tn−2 . . . t1 Dt = (−1)t−1 q −t−k+1 Dt + E1
= (−1)t−1 q −
(6.8.3)
n+t−2
2
Dt + E1
where E1 is a linear combination of diagrams with < t through strings. The same computation
shows that
tn−1 tn−2 . . . t1 (Ḋt ) = (−1)t−1 q −t−k+1 (Ḋt ) + E2
(6.8.3)
= (−1)t−1 q −
n+t−2
2
(Ḋt ) + E2
where E2 is a linear combination of diagrams with < t through strings. It follows that
βtn−1 tn−2 . . . t1 t0 Dt = βtn−1 tn−2 . . . t1 t0 Dt = −(Ḋt ) − Q−1 Dt
n+t−2
n+t−2
= β(−1)t q − 2 (Ḋt ) + Q−1 q − 2 Dt + E
= zt Dt + Qzt (Ḋt ) + E,
where E is a combination of diagrams with < t through strings.
✷
We now turn to the case t = 0.
Recall that the diagrams D0 , D0 have been defined above, before the statement of Lemma (6.4).
The following relations are easily verified.
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(D0 )∗ D0 = (D0 )∗ D0 = δq id0 ,
(6.9)
and
Q
q
+
(c0 D0 )∗ D0 = (D0 )∗ c0 D0 = κ id0 =
id0
q
Q
where id0 is the empty diagram 0 → 0, in either the category Ta or TB. Note that although
in the discussion above it is always assumed that n 1, from the point of view of diagram
algebras, the point 0 is an object in both categories Ta and TB. The corresponding algebras are
TLB 0 (q, Q) = R. id0 ∼
= R and T0a (q) = Rτ0 ∼
= RN.
(6.10) L EMMA. – Let γ ∈ X0 be a finite diagram. Then
θ(γτ0 ) = βδq γ(D0 )∗ t0 t1 . . . tn−1 D0 .
Proof. – First, observe that in the category Ta , we have the following relation:
τ0 = D0∗ τn D0 = (D0 )∗ τn D0 .
(6.10.1)
It follows that for finite γ ∈ X0 ,
θ(γτ0 ) = gβ (γτ0 D0∗ )D0
= gβ (γ(D0 )∗ τn D0 D0∗ )D0
by (6.10.1)
= gβ (γ(D0 )∗ τn )gβ (D0 D0∗ )D0
= gβ (γ(D0 )∗ τn )D0 D0∗ D0 since
= δq gβ (γ(D0 )∗ τn )D0
D0 D0∗ is finite
by (6.9)
= δq gβ (γ(D0 )∗ )gβ (τn )D0
= δq βγ(D0 )∗ t0 t1 . . . tn−1 D0
since γ(D0 )∗ is finite
✷
(6.11) L EMMA. – We have
t1 t2 . . . tn−1 D0 = q −
n−4
2
D0 .
Proof. – Denote by D0 (n − 2) the diagram from 0 to n which looks like D0 : 0 → n − 2, with
the rightmost top vertices joined by a horizontal arc. Then clearly cn−1 D0 = D0 (n − 2). Further,
cn−2 D0 = δq D0 , and cn−2 D0 (n − 2) = D0 . Thus
tn−1 D0 = − cn−1 + q −1 D0 = − D0 (n − 2) + q −1 D0 .
Hence
tn−2 tn−1 D0 = cn−2 + q −1 D0 (n − 2) + q −1 D0
= D0 + q −1 δq D0 + q −1 D0 (n − 2) + q −2 D0
= q −1 D0 (n − 2).
Repeating this argument
n−2
2
times, we obtain
t1 t2 . . . tn−1 D0 = q −
n−2
2
t1 D 0 .
But c1 D0 = δq D0 , whence t1 D0 = −(δq + q −1 )D0 = qD0 , and the lemma follows. ✷
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Proof of Theorem (6.1). – We show first that for any element γ ∈ Xt (t 0), we have
θ(γτt ) = χz θ(γ),
(6.1.1)
where z = zt = (−1)t βQ−1 q − 2 (n+t−2) and
1
χz =
(6.1.2)
z
z + z −1
if t > 0,
if t = 0.
By Lemma (6.6)(ii), it suffices to prove (6.1.1) for any finite diagram γ ∈ Xt .
We prove (6.1.1) first for t > 0. Recall that if E is a positive morphism (diagram) in TB,
Ė is obtained from E by marking the leftmost through string. Then observe that in the above
˙ t , and if we regard Wt (n) as a quotient of the bimodule
notation, Dt∗ Dt = idt , while Ḋt∗ Dt = id
˙ t = 0. Moreover, if E ∈ TB
HomTB(t, n) in the obvious way, then in Wt (n), γ idt = γ, while γ id
is a linear combination of diagrams with fewer that t through strings, then γE = 0 by definition.
Hence
θ(γτt ) = gβ (γτt Dt∗ )Dt by (6.5.1)
= gβ γ(Dt )∗ τn Dt since τt Dt∗ τn−1 = (Dt )∗
= gβ γ(Dt )∗ gβ (τn )Dt
= βγ(Dt )∗ t0 t1 . . . tn−1 Dt
by (6.5.1) again, and (5.13)
= γ(βtn−1 tn−2 . . . t1 t0 Dt )∗ Dt since ∗ is an anti-involution
∗
= γχz Dt + Q(Ḋt ) + E Dt by (6.8)
= χz θ(γ),
which proves (6.1.1) for t > 0. Now take t = 0 and repeat the above computation using
Lemmas (6.10), (6.11) instead of (6.8). One obtains
θ(γτ0 ) = βδq γ(D0 )∗ t0 t1 . . . tn−1 D0
= βδq γ(D0 )∗ t0 q
by (6.10)
− n−4
2
D0 by (6.11)
= −βδq q
c0 + Q−1 D0
n−4 = −βδq q − 2 γ κ id0 +Q−1 δq id0
n−4
Q
−1 −1
−Q q
=−
βδq q − 2 γ
q
n−2
= − Q − Q−1 βq − 2 θ(γ) by Corollary (6.7)(ii).
− n−4
2
n−2
γ(D0 )∗
But z0 = βQ−1 q − 2 , and since β 2 = −q n−2 , β −1 = −q −(n−2) β. It follows easily that
θ(γτ0 ) = (z0 + z0−1 )θ(γ), which completes the proof of (6.1.1).
It follows from (6.1.1) (cf. (6.7)) that θ induces a homomorphism which we also denote by
θ : Wt,z (n) → Zt , where z = zt . Note that Wt,z (n) and Zt are free R-modules of the same rank.
To complete the proof of the theorem, we discuss the cases t > 0 and t = 0 separately.
If t > 0, then by (6.7), θ is surjective. It follows, since Wt,z (n) and Zt are free R-modules
of the same rank, that θ is an isomorphism. If t = 0, the same argument (using (6.7)) shows
that θ defines an isomorphism Wt,z (n) → δq Zt . But since δq is not a zero-divisor, δZt ∼
= Zt as
Tna (q)-modules, which completes the proof. ✷
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Theorem (6.1) shows in particular that any cell module Wt,z (n) for the R-algebra Tna (q) may
be realised as the pullback of a cell module of the algebra TLB n (q, Q).
(6.12) C OROLLARY. – For any (relevant) pair t, z, the cell module Wt,z (n) may be realised
1
∗
as gβ,Q
Wt (n), where Q = Q(t, z) = (−1)t βz −1 q − 2 (n+t−2) .
Recall from the beginning of this section that we also have cell modules Wt (n) for
TLB n (q, Q), for t < 0. We next identify their lift to Tna (q) via gβ,Q .
(6.13) T HEOREM. – Suppose δq is not a zero divisor in R and let t satisfy 0 t n,
t ≡ n (mod 2). Then the inflation
gβ ∗ W−t (n) ∼
= Wt,q−t z−1 (n) = Wt,−Q2 zt (n)
t
where zt = (−1)t βQ−1 q − 2 (n+t−2) as in (6.1).
1
Proof. – The case t = 0 has been treated above. Note that z0 = βQ−1 q − 2 (n−2) , so that
2
z0 = −Q−2 . Thus −Q2 z0 = z0−1 , and since W0,z (n) = W0,z−1 (n), the case t = 0 follows from
Theorem (6.1). We therefore take t > 0.
First observe that in analogy with (6.5), we have a homomorphism of Tna (q)-modules
−
θ : Xt → gβ ∗ W−t (n) given by
1
(6.13.1)
θ− (γ) = gβ (γDt∗ )(Ḋt ),
where γ ∈ Xt , and Dt∗ and Ḋt = Dt ◦ c0 (t) are the diagrams defined in the proof of (6.3),
regarded as diagrams in Ta and TB respectively (see immediately preceding (6.8)) that for a
diagram E with unmarked through strings, Ė denotes the same diagram with the leftmost through
string marked. One verifies easily that for a finite diagram γ : t → n(∈ Xt ) we have
θ− (γ) = (γ̇).
(6.13.2)
In accordance with the strategy of the proof of Theorem (6.1), to prove the theorem it will suffice
to show that for any finite diagram γ ∈ Xt , we have
(6.13.3)
θ− (γτt ) = q −t zt−1 θ− (γ) = q −t zt−1 (γ̇).
Arguing as in the proof of (6.1), we have
θ− (γτt ) = gβ (γτt Dt∗ )(Ḋt )
= gβ (γ(Dt )∗ τn )(Ḋt ) since τt Dt∗ τn−1 = (Dt )∗
(6.13.4)
= γ(Dt )∗ gβ (τn )(Ḋt )
= γ(Dt )∗ βt0 t1 . . . tn−1 (Ḋt )
= γ(βtn−1 tn−2 . . . t1 t0 Dt )∗ (Ḋt ) since ∗ is an anti-involution
∗
= γzt Dt + Q(Ḋt ) + E (Ḋt ) by (6.8),
where E is a linear combination of diagrams with fewer than t through strings.
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Now a straightforward computation shows that
˙ t ) and D∗ (Ḋ ) = id
˙ t.
(Ḋt∗ )(Ḋt ) = δQ (id
t
t
Hence from (6.13.4) we have
∗
θ− (γτt ) = γzt Dt + Q(Ḋt ) + E (Ḋt )
˙ t )(1 + QδQ )
= zt γ(id
˙ t)
= −Q2 zt γ(id
= −Q2 zt γ̇.
But zt2 = −q −t Q−2 , whence q −t zt−1 = −Q2 zt , and (6.13.3) follows. This completes the proof
of Theorem (6.13). ✷
Notice that for any t, if zt = (−1)t βQ−1 q − 2 (n+t−2) , then z−t = q t zt . Hence in particular if
−1
t 0, q t z−t
= zt−1 . Hence Theorem (6.13) may be stated as follows.
1
(6.14) C OROLLARY. – Suppose t ∈ T and t 0. Then the inflation gβ∗ Wt (n) ∼
= W−t,z−1 (n).
t
We may combine and generalise the statements of Theorems (6.1) and (6.13) as follows. Note
that the condition on δq is absent from the statement below.
(6.15) C OROLLARY. – Suppose R is any commutative ring and suppose that q, Q are
elements of R× . Let β ∈ R× satisfy β 2 = −q n−2 and let gβ : Tna (q) → TLB n (q, Q) be the
surjection defined in (5.11). For t ∈ Z such that |t| n and t ≡ n (mod 2), define εt := t/|t| for
t = 0, and εt = 1 if t = 0.
Then the inflation gβ∗ Wt (n) of the cell module Wt (n) of TLB n (q, Q) is isomorphic to the cell
module W|t|,ztεt (n) of Tna (q), where zt is as defined in the statement of (6.1).
Proof. – Replace R in the statement of Theorems (6.1) or (6.13) by the ring R[q, q−1 ] of
Laurent polynomials in the indeterminate q and replace q ∈ R by q ∈ R[q, q−1 ]. Then all
algebras and modules may be considered over R[q, q−1 ], and the hypotheses of Theorems (6.1)
or (6.13) apply and we deduce that there is a (unique) isomorphism of R[q, q−1 ]-modules:
gβ∗ Wt (n) → W|t|,ztεt (n)
which takes a finite diagram in Wt (n) to the same diagram in W|t|,ztεt (n). But R is a module
over R[q, q−1 ] via the homomorphism which takes q to q. Tensoring the above isomorphism
with R provides the required isomorphism. ✷
7. Eigenvalues of the translation elements Xi
na (q) → Tna (q) and that we
Recall that we have a 1-parameter family of surjections ψα : H
∗
wish to study the inflations ψα Wt,z (n). Now by Corollary (6.12), Wt,z (n) may be realised as
a pullback via gβ = gβ,Q of some cell module Wt (n) (for t 0) of TLB n (q, Q). But by the
commutativity of the diagram (5.12),
(7.1)
∗
Wt (n),
ψα∗ Wt,z (n) ∼
= ψα∗ gβ∗ Wt (n) ∼
= ξQ,µ
where α, β and µ are related by β 2 = −q n−2 and µα−1 = β.
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∗
Thus we study the structure of the pullbacks ψα∗ Wt,z (n) by means of the pullbacks ξQ,µ
Wt (n).
For this we shall require some easy facts concerning the submodule structure of Wt (n) on
restriction to TLB n−1 (q, Q). For 1 i n, we regard TLB i (q, Q) as the subalgebra generated
by {c0 , c1 , . . . , ci−1 }; in terms of diagrams, it is the subalgebra spanned by diagrams with the
rightmost n − i top and bottom vertices joined by unmarked vertical through strings. Throughout
this section we assume that δq is not a divisor of zero in R, since we shall require Theorems (6.1)
and (6.13).
(7.2) P ROPOSITION. – Let t ∈ Z, 0 |t| n, n + t ∈ 2Z.
(i) If t 0, we have a short exact sequence
(7.2.1)
TLB (q,Q)
0 → Wt−1 (n − 1) → ResTLB nn−1 (q,Q) Wt (n) → Wt+1 (n − 1) → 0.
(ii) If t < 0, we have a short exact sequence
(7.2.2)
TLB (q,Q)
0 → Wt+1 (n − 1) → ResTLB nn−1 (q,Q) Wt (n) → Wt−1 (n − 1) → 0.
Here we adopt the convention that Wj (k) = 0 if |j| > k.
Proof. – (i) If t > 0, the diagrams in Wt (n) in which the rightmost top and bottom vertices
are joined span a TLB n−1 (q, Q)-submodule which is clearly isomorphic to Wt−1 (n − 1).
The quotient module is spanned by the images of the other diagrams, which all have the
property that the top right vertex is joined to another top vertex. Define a homomorphism
Wt (n)/Wt−1 (n − 1) → Wt+1 (n − 1) by sending a basis diagram ν of Wt (n) \ Wt−1 (n − 1)
to the diagram t + 1 → n − 1 obtained by moving the top right vertex to the bottom right without
changing any arcs. One checks easily that this is an isomorphism of TLB n−1 (q, Q)-modules.
If t = 0, the diagrams in Wt (n) = W0 (n) such that the arc from the rightmost vertex is
marked span a TLB n−1 (q, Q)-submodule which is clearly isomorphic to W−1 (n − 1) (imagine
the rightmost vertex as the bottom vertex in a marked diagram 1 → n − 1). The quotient
W0 (n)/W−1 (n − 1) is spanned by the other diagrams, and by a similar argument is isomorphic
as TLB n−1 (q, Q)-module to W1 (n − 1).
(ii) If t < 0, the same construction as in (i) (but with care if t = −1, in which case we
take the submodule spanned by diagrams with the rightmost top and bottom vertices joined
by a marked string) provides a Wt−1 (n − 1)-submodule of Wt (n) which is isomorphic to
W−(|t|−1) (n − 1) = Wt+1 (n − 1), with corresponding quotient Wt−1 (n − 1). ✷
(7.3) C OROLLARY. – Let t ∈ Z, 0 |t| n, n + t ∈ 2Z and suppose t 0. There is a
filtration of Wt (n) by R-submodules W (i) , i = n, n − 1, n − 2, . . . , 1, as in (7.3.1) below.
Wt (n) ⊃ Wt−1 (n − 1) ⊃ · · · ⊃ W0 (n − t) ⊃ W−1 (n − t − 1) ⊃ W0 (n − t − 2)
(7.3.1)
Thus
⊃ W−1 (n − t − 3) ⊃ · · · ⊃ W0 (2) ⊃ W−1 (1).


 Wt−n+i (i)
(i) ∼
W = W0 (i)


W−1 (i)
For each i = 1, 2, 3, . . ., n, W
(i)
if n − t i n,
if i is even and 0 i n − t,
if i is odd and 0 i n − t.
is a TLB i (q, Q)-submodule of Wt (n).
Proof. – This is immediate by repeated application of (7.2), which also gives an explicit
description of the W (i) in terms of the diagrams of Wt (n) which span them. For the reader’s
convenience, we give this description here. For 0 i n − t, W (i) is the R-submodule spanned
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by diagrams with the rightmost n − i top vertices joined to the rightmost bottom vertices by
through strings. The “remaining” part of the diagram may then be thought of as an element of
Wt−n+i (i). In particular W (n−t) ∼
= W0 (n − t).
For i = n − t − 1, we take W (i) to be the R submodule spanned by diagrams in W (n−t) where
the (n − t)th top vertex is on a marked arc. By thinking of this (n − t)th vertex as moved to the
bottom, it is clear that W (n−t−1) ∼
= W−1 (n − t − 1). The next term W (n−t−2) is spanned by
(n−t−1)
such that the top nodes n − t and n − t − 1 are joined (with a marked
diagrams in W
arc). Clearly W (n−t−2) ∼
= W0 (n − t − 2), and we may now repeat the above construction until
the sequence terminates. ✷
To illustrate the construction in (7.3), observe that the diagram Et = Et,n below lies in each
of the submodules W (i) .
a (q) on the
We shall now determine the eigenvalues of the translation elements Xi ∈ H
n
∗
Wt (n) (cf. (7.1)).
inflation ξQ,µ
a (q) acts on the module
(7.4) P ROPOSITION. – The element X1 X2 . . . Xn ∈ H
n
∗
Wt (n)
M = ξQ,µ
as multiplication by the scalar
(7.4.1)
h(n, t) = (−1)
n+t
2
µn Q−t q 2 (2n−n
1
2
−t2 )
.
∗
Wt (n) ∼
Proof. – First take t 0. By (7.1), ξQ,µ
= ψα∗ Wt,z (n), where z = zt . But by (3.6.1),
X1 X2 . . . Xn = V n , and by (5.13), ψα (V n ) = αn τ n ∈ Tna (q). Moreover τ n is central in
∗
Wt (n)
Tna (q) and acts on Wt,z (n) as multiplication by z t . Hence X1 X2 . . . Xn acts on ξQ,µ
as multiplication by the scalar
t
1
αn ztt = αn (−1)t βQ−1 q − 2 (n+t−2) ,
which after simplification using the relations β 2 = −q n−2 and β = α−1 µ is easily shown to be
equal to h(n, t).
If t < 0, the proof is the same, but we use Theorem (6.13) instead of (6.1). ✷
For the remainder of this section, we take t 0.
∗
na (q)-module on Wt (n), and for
Wt (n) defines a structure of H
The inflation ξQ,µ
(i)
i = 1, 2, . . . , n, the submodule W
of the filtration (7.3.1) is stable under the action of the
a (q) of H
a (q) which
subalgebra TLB i (q, Q) which is the image under ξµ,Q of the subalgebra H
n
i
(i)
a
is generated by {X1 , T1 , . . . , Ti−1 }. We shall refer to the submodules W as Hi (q)-modules,
a (q) contains the elements
with the understanding that the action is induced by ξµ,Q . Now H
i
{X1 , X2 , . . . , Xi } and it follows from (7.4) that
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a (q) acts on the submodule W (i) of
(7.5) C OROLLARY. – The element X1 X2 . . . Xi ∈ H
n
as multiplication by the scalar βi , where
∗
ξQ,µ
Wt (n)

t−n+2i
i n−t−i 12 (2i−i2 −(t−n+i)2 )
q

 (−1) 2 µ Q
2
i
1
i
(2i−i
)
βi = (−1) 2 µ q 2


i−1
2
1
(−1) 2 µi Qq 2 (2i−i −1)
if n − t i n,
if i is even and 1 i n − t,
if i is odd and 1 i n − t.
Proof. – Using the identification of W (i) as a TLB i (q, Q)-module given in (7.3), it follows
from (7.4) that X1 X2 . . . Xi acts as a scalar βi on W (i) , and that in the three cases n − t i n,
1 i n − t and i even and 1 i n − t and i odd, the value of βi is respectively given by
h(i, t − n + i), h(i, 0) and h(i, −1). The formulae in the statement are the result. ✷
We shall determine the action of the Xi using the fact that
Xi = (X1 X2 . . . Xi )(X1 X2 . . . Xi−1 )−1 .
(7.6) C OROLLARY. – In the filtration of (7.3)
Wt (n) = W (n) ⊃ · · · ⊃ W (i) ⊃ W (i−1) ⊃ · · · ⊃ W (1) ⊃ 0,
a (q)-module, we have, for any pair of integers i, j such that
where W (i) is thought of as a H
i
1 j < i n that Xi acts on W (j) as multiplication by γi , where γi is given by

−1 n+2−t−2i

 −µQ q
γi = −µQ−1 q 2−i


µQq 1−i
if n − t i n,
if i is even and 1 i n − t,
if i is odd and 1 i n − t.
Proof. – Since W (j) is a submodule of W (i−1) , it will suffice to show that Xi acts on W (i−1)
as multiplication by γi . But W (i−1) is a submodule of W (i) , and hence by (7.5), the elements
Yi = X1 X2 . . . Xi and Yi−1 = X1 X2 . . . Xi−1 act as the scalars βi , βi−1 respectively on W (i−1)
−1
−1
(here we take Y0 = 1). It follows that Xi = Yi Yi−1
acts on W (i−1) as the scalar γi = βi βi−1
,
where β0 = 1. An easy calculation now completes the proof. ✷
(7.7) P ROPOSITION. – There is an ordering of the diagrams which form an R-basis of
Wt (n) which is compatible with the filtration (7.3) and with respect to which each element Xi
(i = 1, 2, . . . , n) has upper triangular matrix.
Proof. – We use induction on n; for n = 1 the result is trivial. Assume the result true for
Wt−1 (n − 1) and Wt+1 (n − 1). Then there is an ordering of the diagrams in W (n) which is
compatible with the filtration W (n) ⊃ W (n−1) ⊃ · · · ⊃ 0 such that X1 , . . . , Xn−1 act in upper
triangular fashion. This is because by induction we have such an ordering of the diagrams
in W (n−1) , while W (n) /W (n−1) ∼
= Wt−1 (n − 1) or W (n) /W (n−1) ∼
= Wt+1 (n − 1), whence
again by induction, there is an appropriate ordering of the other diagrams in Wt (n). But by (7.4),
−1
Yn := X1 X2 . . . Xn acts as a scalar on Wt (n), whence Xn = Yn X1−1 X2−1 . . . Xn−1
is also upper
triangular. ✷
na (q) generated by
(7.8) D EFINITION. – (i) We denote by U (n) the subalgebra of H
±1
±1
{X1 , . . . , Xn }. By (3.3) this is isomorphic to the ring R[X1 , . . . , Xn ] of Laurent polynomials
in the Xi .
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J.J. GRAHAM AND G.I. LEHRER
(ii) A character χ : U (n) → R× is the linear extension to U (n) of a multiplicative
homomorphism χ : U (n)× → R× , where U (n)× is the group generated by the Xi , i = 1, . . . , n
(which is isomorphic to Zn ). Such a character is clearly determined by the images of the Xi and
hence the group of characters may be identified with (R× )n .
(iii) If χ : U (n) → R× is a character, write σ(χ) = (χ(X1 ), . . . , χ(Xn )) ∈ (R× )n and |σ(χ)|
for the multiset (i.e. set with multiplicities) {χ(X1 ), . . . , χ(Xn )}. We sometimes identify σ(χ)
with χ and refer to the sequence σ(χ) as a character.
Now any diagram D ∈ Wt (n) may be considered as an element of an ordered basis of Wt (n)
as in (7.7). Hence D defines a character χD of U (n) by
(7.9)
Xi D ∈ χD (Xi )D +
RE,
E∈Wt (n),E=D
where the sum is over diagrams in Wt (n) which are distinct from (and ordered lower than) D.
(7.10) T HEOREM. – Assume t 0. For any diagram D ∈ Wt (n), the character χD satisfies
|σ(χD )| = {γ1 , γ2 , . . . , γn }, where the γi are as in (7.6), i.e.,
{γ1 , . . . , γn } = µQ, µQq −2 , µQq −4 , . . . , µQq −(n−t−2)
∪ −µQ−1 , −µQ−1 q −2 , −µQ−1 q −4 , . . . , −µQ−1 q −(n+t−2) ,
where the union is in the sense of multisets.
Proof. – This will be by induction on n. For n = 1 the result is trivial. Let D be any
diagram in W (n) . Then there is a unique index i such that D ∈ W (i) \ W (i−1) , where
. . . W (i) ⊃ W (i−1) . . . is the filtration (7.3) of Wt (n). If i < n, it follows by induction on n that
−1
{χD (X1 ), . . . , χD (Xi )} = {γj = βj βj−1
| 1 j i}. Moreover for j > i, it follows from (7.6)
that Xj D = γj D, since Xj acts as γj id on W (i) . Thus the assertion (7.10) is true if i < n. Hence
we consider the case i = n.
If t = n, then by (7.2.1), W (n) /W (n−1) ∼
= Wt+1 (n − 1) as TLB n−1 (q, Q)-module, and
hence
X1 X2 . . . Xn−1 D ≡ h(n − 1, t + 1)D (mod W (n−1) ).
Hence χD (Xn ) = h(n, t)h(n − 1, t + 1)−1 = µQq −(n−t−2) . The set {χD (Xi ) | 1 i n − 1}
may be determined by thinking of the Xi , 1 i n − 1 as acting on the image of D in
W (n) /W (n−1) ∼
= Wt+1 (n − 1). Then by induction we see that {χD (Xi ) | 1 i n − 1} is
the set in the statement of (7.1), with (n, t) replaced by (n − 1, t + 1), i.e.
χD (Xi ) | 1 i n − 1 = {γ1 , . . . , γn } \ µQq −(n−t−2) ,
and hence {χD (Xi ) | 1 i n − 1} is as stated in (7.1). This proves (7.10) for the case t = n.
If t = n, Wt (n) ∼
= Wt−1 (n − 1) as TLB n−1 (q, Q)-module, whence
χD (Xi ) | 1 i n − 1 = {γ1 , . . . , γn } \ −µQ−1 q −(2n−2) .
na (q) acts on the module Wn (n) as multiplication by the scalar
But X1 X2 . . . Xn ∈ H
h(n, n) = (−1)n µn Q−n q (n−n ) .
2
Hence Xn = −µQ−1 q −(2n−2) id on Wn (n), which completes the proof for the case t = n.
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✷
DIAGRAM ALGEBRAS AND DECOMPOSITION NUMBERS
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8. The case of generic parameters
∗
In this section we prove that the inflations ψα∗ Wt,z (n) ∼
Wt (n) are representations of
= ξQ,µ
na (q) which are induced from parabolic subalgebras, provided that the pair of parameters q, Q
H
is “generic”. First we make precise the meaning of this term.
(8.1) D EFINITION. – We say that the pair (q, Q) of invertible elements is generic in the
integral domain R if q is not a root of unity and there is no solution m ∈ Z for the equation
q m = ±Q2 .
(8.2) P ROPOSITION. – Suppose that the pair (q, Q) is generic.
(i) The elements of the multiset {γ1 , . . . , γn } of (7.10) are distinct, i.e. have multiplicity one.
Moreover if χD (Xi ) = χD (Xj )q −2 , then j < i.
(ii) The characters χD are distinct; i.e. if χD1 = χD2 for diagrams D1 , D2 ∈ Wt (n), then
D1 = D2 .
Proof. – Observe first that by (7.6), the sequence (γ1 , . . . , γn ) is given by
µQ, −µQ−1 , µQq −2 , −µQ−1 q −2 , . . . , µQq −(n−t−2) , −µQ−1 q −(n−t−2) ,
(8.2.1)
− µQ−1 q −(n−t) , −µQ−1 q −(n−t+2) , . . . , −µQ−1 q −(n+t−2) .
The first assertion of (i) is clear. For the second assertion, we use the argument in the proof
of (7.10). If D ∈ W (i) \ W (i−1) , then
(i)
(i)
(i)
σ(χD ) = χD (X1 ), χD (X2 ), . . . , χD (Xi ), γi+1 , . . . , γn ,
where χD is the character of U (i) = R[X1±1 , . . . , Xi±1 ] on W (i) which corresponds to D ∈
W (i) . Inspection of the sequence (8.2.1) (and induction on n) now yields the second assertion if
i < n.
If i = n, the argument of (i) shows that
(i)
σ(χD ) = χD (X1 ), χD (X2 ), . . . , χD (Xn−1 ), µQq −(n−t−2) ,
where χD is the character of U (n − 1) corresponding to D ∈ W (n) /W (n−1) ∼
= Wt+1 (n − 1).
Again by induction, the assertion is true for this sequence, whence the result. This proves (i).
The proof of (ii) is also by induction on n. The result is trivial for n = 1. Let D1 , D2 be distinct
diagrams in W = W (n) . If D1 , D2 are either both in W (n−1) or both in W (n) \ W (n−1) , the
result is immediate by induction. Thus we may take D1 ∈ W (n−1) and D2 ∈ W (n) \ W (n−1) .
But then χD1 (X1 . . . Xn−1 ) = χD2 (X1 . . . Xn−1 ) by Theorem (7.4), since in the generic case
h(n − 1, t − 1) = h(n − 1, t + 1). This completes the proof of (8.2). ✷
(8.3) C OROLLARY. – Assume that the pair (q, Q) is generic (see (8.1)). Define sequences
Γ1 , Γ2 as in (8.3.1) below.
(8.3.1)
Γ1 = µQ, µQq −2 , µQq −4 , . . . , µQq −(n−t−2) ,
Γ2 = −µQ−1 , −µQ−1 q −2 , −µQ−1 q −4 , . . . , −µQ−1 q −(n+t−2) .
Then Γ1 , Γ2 are disjoint and have cardinalities k = n−t
2 , n − k respectively. The set of characters
(cf. Definition (7.8)(iii)) {σ(χD ) | D a diagram in Wt (n)} coincides with the set of all orderings
of Γ := Γ1 Γ2 in which Γ1 , Γ2 appear in the given order.
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Proof. – By (7.10), each character χD corresponds to some ordering of Γ. In the generic case,
the elements of Γ are distinct, whence the corresponding ordering is unique. But by the second
assertion in Proposition (8.2)(i), in any ordering corresponding to a character
χD , Γ1 and Γ2
n
, which coincides with
appear in the given order. But the number of such orderings of Γ is n−t
2
the number of distinct diagrams D ∈ Wt (n) (see the paragraph preceding (6.1)). The assertion
follows. ✷
(8.4) C OROLLARY. – Suppose that R is any integral domain and let (q, Q) be generic in
R. There is a non-zero element ∆ ∈ R with the following property. Let R∆ be the localisation
na (q)-module ξ ∗ Wt (n) and let M∆ := R∆ ⊗R M . Then
R∆ = R[∆−1 ]. Write M for the H
Q,µ
M∆ has a unique decomposition M∆ ∼
= D M∆ (D), where the sum is over the diagrams in
Wt (n) and M∆ (D) is a free R∆ -submodule of rank one, which is invariant under U (n)∆ =
R∆ [X1±1 , . . . , Xn±1 ], and on which U (n)∆ acts via the character χD . Any simultaneous
eigenvector E ∈ M∆ for the Xi corresponds to one of the characters χD of U (n).
Proof. – By the first statement in (8.2)(i), the characters χD are distinct. Hence there is an
element Y ∈ U (n) whose set {χD (Y ) | D a diagram in Wt (n)} of eigenvalues (recall that the
elements of U (n) act in triangular fashion on Wt,z (n) and the χD (Y ) are the diagonal elements)
is distinct. It is easy to show using elementary linear algebra that there is a set of elements
xD ∈ M such that Y xD = χD (Y ) (for each diagram D). If ∆ is the (non-zero) determinant of
the matrix formed by the coefficients of the xD with respect to an R-basis of M , it is clear that
M∆ is the direct sum of the rank one submodules R∆ xD , which are each invariant under Y .
Since Y has distinct eigenvalues, any linear transformation of M∆ which commutes with Y is
diagonal with respect to this eigenspace decomposition, whence we deduce that the eigenspaces
of Y are invariant under U (n).
Finally, if E is a simultaneous eigenvector of the Xi , it is an eigenvector of Y , whence
E ∈ M∆ xD for some diagram D. This proves the last statement. ✷
Note that if R is a field in (8.4), the statement may be simplified by eliminating any reference
to ∆.
We shall refer to an ordering of Γ satisfying the condition of (8.3) as permissible. Note that
one permissible ordering is
(Γ) = −µQ−1 , −µQ−1 q −2 , . . . , −µQ−1 q −(n+t−2) , µQ, µQq −2, . . . , µQq −(n−t−2) .
We write (Γ2 , Γ1 ) = (δ1 , . . . , δn ) for this ordering.
(8.5) C OROLLARY. – With notation as in (8.3), let R be a field. Then there is a vector E ∈ M
which is a simultaneous eigenvector of X1 , X2 , . . . , Xn such that χE = (δ1 , . . . , δn ).
This is clear from (8.4).
We shall require the following elementary result from linear algebra.
(8.6) L EMMA. – Let V be a vector space over a field R. Let A, B be commuting linear
transformations of V and let v ∈ V satisfy
(A + B)v = (a + b)v
and ABv = abv
for some a, b ∈ R.
Assume that Av ∈
/ Rv. Then
(i) The plane Π = v, Av = v, Bv is stable under the semigroup generated by A, B.
(ii) The eigenvalues of A, B on Π are each {a, b}.
(iii) If v1 = bv − Av, v2 = av − Av then Av1 = av1 , Bv1 = bv1 , Av2 = bv2 and Bv2 = av2 .
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The proof is easy.
(8.7) P ROPOSITION. – Let E ∈ M be the eigenvector of (8.5). Then Tj E = −q −1 E for
j = n+t
2 , j = 1, 2, . . . , n − 1.
Proof. – Take j ∈ Z, 1 j n − 1, j = n+t
2 , and let v = Tj E. Then since Xi commutes with
Tj unless i = j or i = j + 1 (see (2.6.1) or Proposition (3.3)(iv)), we have
(8.7.1)
X i v = δi v
if i = j, j + 1.
Moreover since Tj commutes with Xj Xj+1 and Xj + Xj+1 (by the Bernstein relation (3.3)(iv)),
we have Xj Xj+1 v = δj δj+1 v and (Xj + Xj+1 )v = (δj + δj+1 )v.
Observe that Xj v = Xj Tj E = Tj−1 Xj+1 E = δj+1 Tj−1 E. But since Tj−1 = Tj + q −1 − q, we
have
(8.7.2)
Xj v = δj+1 Tj E + q −1 − q E = δj+1 v + q −1 − q E .
Suppose Xj v is not a scalar multiple of v. Then we may apply (8.6) with A = Xj , B = Xj+1 ,
a = δj and b = δj+1 . By (8.6) the vector v2 = δj v − Xj v is a simultaneous eigenvector for
Xj , Xj+1 with respective eigenvalues δj+1 , δj . Moreover by (8.7.1), v2 is also an eigenvector for
each Xi (i = j, j + 1) with corresponding eigenvalue δi . Thus v2 corresponds to some character
χD of U (n) by the last statement in (8.4). But our choice of j ensures that δj+1 = δj q −2 , which
by the second statement in (8.2)(i) is impossible. It follows that Xj v ∈ Rv. Hence by (8.7.2),
Tj E ∈ RE. By the quadratic relation for Tj , we therefore have Tj E = qE or Tj E = −q −1 E;
but Xj+1 E = Tj Xj Tj E = q −2 Xj E, whence Tj E = −q −1 E as stated. ✷
9. Induced modules and standard modules
We shall henceforth assume that R is a field and all modules and algebras will be R-modules
and R-algebras.
∗
We shall identify M = ξQ,µ
Wt (n) as a module induced from a one-dimensional module
of a parabolic subalgebra of Hna (q). Such modules were introduced by Rogawski [28]. The
a (q)
parabolic subalgebras are defined as follows. For any subset J ⊂ {1, 2, . . . , n − 1} let H
J
na (q) which is generated by U (n) = R[X ±1 , . . . , Xn±1 ] and {Tj | j ∈ J}.
be the subalgebra of H
1
a (q) to be H
a (q), where J = Jλ ,
For any partition λ = λ1 λ2 · · · λ 0 of n, define H
J
λ
k
defined as follows: j ∈ Jλ ⇔ ∃k ∈ {1, 2, . . . , 4} with n i=1 λi j + 1 > j > k−1
i=1 λi ,
a (q) has finite index equal
where, if k = 0, the smaller sum is interpreted as 0. The subalgebra H
λ
n!
a (q).
to the multinomial coefficient λ1 !...λ
in
H
n
!
a (q)-module on which Tj acts as multiplication by q
Now let Rλ,χ be a one-dimensional H
λ
and U (n) = R[X1±1 , . . . , Xn±1 ] acts via the character χ. Then since Xj+1 = Tj Xj Tj
(9.1)
χ(Xj+1 ) = q 2 χ(Xj )
for each j ∈ Jλ ,
a (q)-module.
and any character satisfying this condition gives rise to a one-dimensional H
λ
A similar definition applies when Jλ is replaced by an arbitrary subset J of {1, . . . , n − 1}.
(9.2) D EFINITION. – Let χ be a character of U (n) = R[X1±1 , . . . , Xn±1 ] which satisfies
a (q)-module Kλ,χ as the induced module
condition (9.1). Define the H
n
na (q) ⊗ a Rλ,χ ,
Kλ,χ = H
(q)
H
λ
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a (q)-module defined above. Then the dimension of Kλ,χ
where Rλ,χ is the one-dimensional H
λ
n!
is λ1 !...λ ! .
a (q), where Jλ
Of course this definition applies slightly more generally to the subalgebras H
J
is replaced by an arbitrary subset J ⊂ {1, 2, . . . , n − 1} and the corresponding module is denoted
by KJ,χ .
We have the following easy characterisation of these induced modules.
a (q)-module which contains a vector e
(9.3) L EMMA. – Suppose K is a finite dimensional H
n
a
which generates K as Hn (q)-module and satisfies the following two conditions for some subset
J ⊂ {1, 2, . . ., n − 1}:
Ti e = qe for i ∈ J
(9.3.1)
and
(9.3.2)
Xi e = χ(Xi )e,
1 i n, for some character χ of U (n).
Then K is a quotient of KJ,χ . In particular, if dim(K) = dim(KJ,χ ), we have equality.
We wish to identify these modules as “standard modules” in the sense of Kazhdan and Lusztig.
Write G = GLn (R) and G = Lie(G). Then U (n) = R[X1±1 , . . . , Xn±1 ] may be thought of as
the coordinate ring of the maximal torus S of G consisting of diagonal elements. Any character χ
of U (n) = R[X1±1 , . . . , Xn±1 ] is therefore given by evaluation at a (generically) unique diagonal
element s ∈ S. It is not difficult to see that in fact the set of pairs (λ, χ), where χ satisfies
condition (9.1), modulo the equivalence defined by isomorphism of the corresponding induced
of equivalence classes of pairs (s, N )
representations Kλ,χ , is thus in bijection with the set P
modulo G, where s ∈ G is semisimple, N ∈ G is nilpotent and s.N = q 2 N (the action of G on G
being the adjoint representation). The correspondence may be realised as follows. Let Jk denote
the k × k (Jordan) matrix with zeros everywhere except on the super-diagonal, where all entries
are 1. Given a partition λ = λ1 · · · λ > 0 of n, there is a unique block diagonal matrix Nλ
whose diagonal (Jordan) blocks have size λi , i = 1, 2, . . . , 4. If R is algebraically closed, then
each pair (s, N ) such that Ad(g) · N = q 2 N is G-conjugate to one where


N = Nλ = 

···
··· 

Jλ1
Jλ2
···
···
Jλ
and s is also block diagonal, of the form


s = sλ = 

···
··· 

sλ1
sλ2
···
···
sλ
where the matrices sλi are diagonal of the form

sλi



=


4e SÉRIE – TOME 36 – 2003 – N◦ 4

ai
ai q
−2



.


ai q −4
.
.
ai q −2(λi −1)
DIAGRAM ALGEBRAS AND DECOMPOSITION NUMBERS
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The matrix sλ depends on the 4 parameters ai ∈ R× , and the two pairs (sλ , Nλ ) and (sλ , Nλ )
are G-conjugate if and only if λ = λ and some parts of λ are equal and the corresponding sλi
are permuted to obtain s .
and pairs (λ, χ) as in (9.2) we shall
In view of the correspondence between pairs (s, N ) ∈ P
use the notation
Kλ,χ = Ks,N .
(9.4)
The character χ is given in terms of the pair (s, N ) by χ(Xi ) = s−1
i , where si is the ith diagonal
entry in si .
Whenever we use the notation (s, N ), we shall assume that R = C unless we specify
otherwise, since we aim to integrate our results with the body of literature which relates to this
case. Now Chriss and Ginzburg [5] and Kazhdan and Lusztig [19], (see [30] for an exposition)
a (q) (of course as a special case of a general
have defined “standard modules” Ms,N for H
n
s
construction) in the case R = C. The space of Ms,N is the Borel–Moore homology H∗ (BN
) of
s
a certain subvariety BN of the flag variety of G. We shall need the following result of Ariki. For
na (q)-module M , [M ] denotes its class in the Grothendieck group of
any finite dimensional H
a
finite dimensional Hn (q)-modules.
(9.5) P ROPOSITION [1, Theorem 3.2, p. 798]. – We have, in the Grothendieck group of finite
na (q)-modules,
dimensional H
a (q) ⊗ a Cλ,χ ,
[Ms,N ] = [Ks,N ] = [Kλ,χ ] = H
n
(q)
H
λ
where N = Nλ and χ(Xi ) = s−1 if s = diag(s1 , . . . , sn ).
for each pair (s, N ) ∈ P,
i
a (q) → H
a (q) which takes Ti to −T −1
Next observe that there is an involution ι : H
n
n
i
−1
(i = 1, . . . , n − 1) and Xj to Xj (j = 1, . . . , n); this follows by noting that the images of the
na (q)-module
Ti and Xj under ι satisfy relations in Proposition (3.3)(iv). It follows that for any H
M , we may define its inflation ι∗ M via ι in the usual way.
(9.6) T HEOREM. – Let R be any field, and suppose the pair (q, z) of invertible elements
of R is generic (see (8.1)). Assume there is an element β ∈ R× such that β 2 = −q n−2 . Let
na (q) → Tna (q) (α ∈ R× ) be the surjection of (5.12) (see also (5.13) and (4.3)(i)) and let
ψα : H
Wt,z (n) be a cell module for the diagram algebra Tna (q), where t ∈ Z, 0 t n, n − t ∈ 2Z
na (q)-module ι∗ ψα∗ Wt,z (n). Then M is isomorphic to the induced
and z ∈ R× . Let M be the H
n−t
module Ksλ ,Nλ (see (9.4)) where λ is the partition ( n+t
2 , 2 ) and sλ is the diagonal matrix
diag(s1 , . . . , sn ), where the si are given by
(s1 , . . . , sn ) = a1 , a1 q −2 , . . . , a1 q −(n+t−2) , a2 , a2 q −2 , . . . , a2 q −(n−t−2) ,
where a1 = (−1)t+1 αzq 2 (n+t−2) and a2 = (−1)t+1 αz −1 q 2 (n−t−2) .
1
1
Proof. – Note first that from diagram (5.12), we see that if α, β and µ are related by
β 2 = −q n−2 and µα−1 = β, then (cf. (7.1))
∗
ψα∗ Wt,z (n) ∼
Wt (n)
= ψα∗ gβ∗ Wt (n) ∼
= ξQ,µ
where Wt (n) is the cell module for TLB n (q, Q) and z = zt (see Theorem (6.1)). But given t, z
∗
and β, (6.12) shows that Wt,z (n) ∼
Wt (n), where
= gβ∗ Wt (n) = gβ,Q
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Q = Q(t, z) = (−1)t βz −1 q − 2 (n+t−2) .
1
Moreover Q(t, z)2 = −z −2 q −t , whence if (q, z) is generic, so is (q, Q).
It follows from (8.5) and (8.7) that there is a vector E ∈ M such that Xi E = δi E for
i = 1, 2, . . . , n, where (δ1 , . . . , δn ) is the sequence in (8.5) and Tj E = −q −1 E for j = 1, 2, . . . ,
−1
n − 1, j = n+t
2 . Hence ι(Xi )E = δi E and ι(Tj )E = qE for each relevant i, j. Since M is
a (q)-module. It follows
irreducible because we are in the generic case, E generates M as H
n
from Lemma (9.3) and from the description of the character χ in (9.4) that M is isomorphic to
n−t
the induced module Ks,Nλ , where λ = ( n+t
2 , 2 ) and s is the diagonal matrix with diagonal
entries
(δ1 , . . . , δn ) = −µQ−1 , −µQ−1 q −2 , . . . , −µQ−1 q −(n+t−2) , µQ,
µQq −2 , . . . , µQq −(n−t−2) ,
where Q = Q(t, z) = (−1)t βz −1 q − 2 (n+t−2) .
1
Using the relations among α, β, z and Q, we see that µQ = (−1)t+1 αz −1 q 2 (n−t−2) and
1
−µQ−1 = (−1)t+1 αzq 2 (n+t−2) . ✷
1
(9.7) Notation. – We fix here some notation for the rest of this work. First, if t ∈ Z,
0 t n, n − t ∈ 2Z, we shall consistently write
k=
(9.7.1)
n−t
.
2
Corresponding to each such t (or k) we have the 2-step partition λt = (n − k, k). Note that
n − k k.
The diagonal matrix in the statement of (9.6) will be denoted sk (a1 , a2 ) or just s(a1 , a2 ), i.e.
(9.7.2)
s(a1 , a2 ) = sk (a1 , a2 ) = diag(s1 , . . . , sn )
= diag a1 , a1 q −2 , . . . , a1 q −(n+t−2) , a2 , a2 q −2 , . . . , a2 q −(n−t−2) .
Recall the definition of the induced module Kλ,χ where λ is a partition of n and χ is a
character of U (n) which satisfies condition (9.1). A special case of this definition is when
λ = λt as above, and χ corresponds to the semisimple element sk (a1 , a2 ) of (9.7.2), i.e.
χ(Xj ) = s−1
j . We denote the corresponding induced module of the statement of (9.6) in this
case by Ks,N = Ks(a1 ,a2 ),Nk , where Nk = Nλt .
We next wish to identify the Grothendieck class of the module M of (9.6) (i.e. its set of
composition factors) in general, i.e. when (q, z) is not necessarily a generic pair. We shall prove
(9.8) T HEOREM. – Let F be a field, and let (q, z) be a pair of invertible elements of F .
√ 2
√ √
√
Assume that there are elements q, −1 ∈ F such that q 2 = q and −1 = −1. Let M be
na (q)-module ι∗ ψα∗ Wt,z (n) as in (9.6), where H
na (q) is now an F -algebra, and α ∈ F × .
the H
a (q)-modules, M is equivalent to the
Then in the Grothendieck group of finite dimensional H
n
induced module Ksλ ,Nλ (see (9.4)) where λ, sλ and Nλ are as in (9.6).
Proof. – Let R0 = F [q] be the ring of polynomials over F in the indeterminate q. Let R be
√
the completion of R0 at the prime ideal P = (q − q), and let L be the quotient field of R.
√
Thus if we write y = q − q ∈ R0 , R is identified with the ring of power series F [[y]], and
L = F [[y]]y = F [[y]][y −1 ]. The ring R is then a complete rank one discrete valuation ring with
residue field F . Under the residue class map R → F , q2 → q.
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a (q2 ) over the ring R, corresponding to the invertible
Consider the affine Hecke algebra H
n
2
2
element q ∈ R. The pair (q , z) is generic in R, and we may therefore interpret Wt,z (n)
as a module for the R-algebra Tna (q2 ). Write W
t,z (n) for this R-module. Let b ∈ R satisfy
b2 = −q2(n−2) . Such an element obviously exists, and under the residue class map b → β ∈ F
such that β 2 = −q −(n−2) .
Write Q = (−1)t bz −1 q−(n+t−2) , and denote R-modules by means of a tilde. Then by
Theorem (6.1), with the obvious extensions of our standard notation,
(9.8.1)
∼ ∗ ∗ ∼ ∗ ψα∗ W
t,z (n) = ψα gb Wt (n) = ξQ,m Wt (n),
where m = αb.
Since the pair (q2 , z) is generic in R, we may invoke Theorem (9.6) to deduce that
(9.8.2)
∼
= L ⊗R ι∗ ψ ∗ W
L ⊗R M
α t,z (n) = L ⊗R Ksλ ,Nλ .
na (q2 )-modules. We shall show how
The isomorphism (9.8.2) is an isomorphism of L ⊗R H
2
to interpret it in terms of L ⊗R TLB n (q , Q)-modules, where TLB n (q2 , Q) is the R-algebra
defined in (5.3), corresponding to the parameters q2 , Q.
na (q2 ) of a TLB n (q2 , Q)-module.
The left side of (9.8.2) is by (9.8.1) the pullback to H
Hence the kernel of ξQ,m acts trivially on the right side of (9.8.2) and it follows easily that
the kernel of ξQ,m acts trivially on the R-module K
sλ ,Nλ , whence the latter may be thought of
as a TLB n (q2 , Q)-module. Thus the isomorphism (9.8.2) may be interpreted as an isomorphism
of L ⊗R TLB n (q2 , Q) modules. Moreover we may think of the space of either side as an Lvector space, in which there are two R-lattices invariant under TLB n (q2 , Q), which correspond
2
and K
respectively to the representations M
sλ ,Nλ of the R-algebra TLB n (q , Q).
2
Now since the pair (q , z) is generic in R, and therefore generic in L, the L ⊗R TLB n (q2 , Q)
module L ⊗R W
t (n) is absolutely irreducible, whence L is a splitting field for
∼
L ⊗R M
= L ⊗R K
sλ ,Nλ .
and K
The reductions mod P of M
sλ ,Nλ are respectively isomorphic to the TLB n (q, Q)modules M and Ksλ ,Nλ , and since F is a splitting field for TLB n (q, Q) (cf. [14, §5]), the triple
(L, R, F ) is a splitting system for TLB n (q2 , Q) in the sense of [3, p. 17]. Moreover since (q2 , z)
is generic in L, L ⊗R TLB n (q2 , Q) is semisimple. It follows from [op. cit., Proposition 1.9.6,
a (q)-modules) M and Ks ,N have the
p. 19], that the TLB n (q, Q) modules (and hence the H
n
λ
λ
same composition factors, counting multiplicity, which is the required statement. ✷
The above result enables us to determine the composition factors of the standard modules
Ms,N for any pair s, N , where N has at most two Jordan blocks (we refer to such N as “twostep nilpotent”) and when q is arbitrary. It has the following consequence.
na (q) be the affine Hecke algebra over the complex numbers C and
(9.9) C OROLLARY. – Let H
a (q)) be the Grothendieck group of finite dimensional H
a (q)-modules. Let λ = (n − k, k)
let Γ(H
n
n
be a 2-step partition of n (2k n), let t = n − 2k and let s be the diagonal matrix of (9.7.2).
na (q) corresponding to the pair
Write Ms,N for the Kazhdan–Lusztig standard module of H
s = sk (a, b), N = Nk (cf. (9.7)). Then
[Ms,N ] = θα∗ Wt,z (n) ,
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a (q)) of an H
a (q)-module V , Wt,z (n) is a cell module for
where [V ] denotes the class in Γ(H
n
n
na (q) → Tna (q) is the surjection ψα of (5.12) composed with
the diagram algebra Tna (q), θα : H
the involution ι of (9.6), and α, z, t satisfy the equations
k=
n−t
,
2
1
a = (−1)n+1 αzq 2 (n+t−2) ,
(9.9.1)
b = (−1)n+1 αz −1 q 2 (n−t−2) .
1
Proof. – This follows easily from (9.5) and (9.8). The former asserts that [Ms,N ] = [Ks,N ],
while the latter asserts that [Ks,N ] = [θα∗ Wt,z (n)]. The formulae for a and b are in the statement
of (9.6), taking account of the fact that t and n have the same parity. ✷
We shall use (9.9) to study the composition factors of the standard modules Ms,N through our
knowledge of those of the cell modules Wt,z (n). First we need to understand the correspondence
between cell modules and standard modules a little better. Assume henceforth that the ground
ring R is C.
(9.10) C OROLLARY. – Given the pair (s(a, b), Nk ) (see (9.7)) with 2k n, there exist
(t, α, z) such that [Ms(a,b),Nk ] = [θα∗ Wt,z (n)]. The parameters (t, α, z) are given by
t = n − 2k,
(9.10.1)
α2 = abq −(n−2) ,
z = (−1)n+1 α−1 aq − 2 (n+t−2) = (−1)n+1 αb−1 q 2 (n−t−2) ,
1
1
where, if t = n (or k = 0), the second equation (for α2 ) is omitted.
∗
Wt,−z (n).
(9.11) L EMMA. – We have ψα∗ Wt,z (n) ∼
= ψ−α
Proof. – For any element β ∈ C such that β 2 = −q n−2 , and α, µ ∈ C such that µ = αβ, we
∗
Wt (n), where
have a commutative diagram (5.12) for any Q ∈ C, and by (6.12), Wt,z (n) = gβ,Q
t
−1 − 12 (n+t−2)
∗
∗ ∗
∗
∼
∼
. Hence ψα Wt,z (n) = ψα gβ,Q Wt (n) = ξQ,µ Wt (n). If we
Q = Q(t, z) = (−1) βz q
replace β by −β and α by −α, µ is unchanged. Thus replacing z by −z leaves Q unchanged,
∗
∗
Wt (n) ∼
Wt,−z (n), as stated. ✷
and it follows that we also have ξQ,µ
= ψ−α
where
(9.12) C OROLLARY. – Let P be the set equivalence classes of pairs (s, N ) ∈ P
N ∈ G is two-step nilpotent, i.e. N ∼ Nk for some k with 0 2k n. Let Ω be the set of
triples (t, α, z) (t ∈ Z, 0 t n, n − t ∈ 2Z; α, z ∈ C× ) and let Ω be the set of equivalence
under the equivalence generated by the relations (t, α, z) ∼ (t, −α, −z),
classes of triples in Ω
−1
(n, α, z) ∼ (n, y zα, y) and (0, α, z) ∼ (0, α, z −1 ). Then (with the obvious abuse of notation)
na (q)-modules Ms,N , (s, N ) ∈ P and θα∗ Wt,z (n) (see (9.9)), (t, α, z) ∈ Ω,
we have well defined H
and there is a bijection f : P → Ω such that if (s, N ) ∈ P corresponds to (t, α, z) ∈ Ω,
[Ms,N ] = [θα∗ Wt,z (n)].
Proof. – The modules Ms,N are the standard modules of Kazhdan–Lusztig. We need to verify
that the isomorphism class of θα∗ Wt,z (n) depends only on the equivalence class of (t, α, z).
But Lemma (9.11) proves this for one type of equivalence, while if t = 0, we observe that
W0,z (n) ∼
= W0,z−1 (n), and if t = n, in the relations (9.9.1) only a occurs, since Nk = N0
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515
has only one Jordan block, and clearly a depends only on αz, so that for any y, z, α ∈ C× ,
θα∗ Wn,z (n) ∼
= θy∗−1 zα Wn,y (n), which completes the verification.
Next, let us define the required bijection f : Ω → P. Given (t, α, z) ∈ Ω, define f (t, α, z) to
be the class of (s(a, b), Nk ), where a, b, k are given by Eqs. (9.9.1). To check that this is well
defined, note that if (t, α, z) is replaced by (t, −α, −z), Eqs. (9.9.1) are unchanged, and therefore
have the same solution. If (0, α, z) is replaced by (0, α, z −1 ), we obtain the solution (s(b, a), N n2 )
of (9.9.1). But the pairs (s(b, a), N n2 ) and (s(a, b), N n2 ) are conjugate under GLn (C). Finally,
the triples (n, α, z) and (n, y −1 zα, y) give the same values of k(= 0) and a, which proves that f
is well defined.
Finally, define f : P → Ω by taking f (s(a, b), Nk ) to be the equivalence class of any triple
(t, α, z) where the parameters satisfy Eqs. (9.10.1). If t = n there are just two such triples, which
are obviously equivalent. If t = n, all the resulting triples are easily seen to be equivalent, since
they have a common value of αz. To check that f is well defined, we note that the pairs
(s(a, b), Nk ) and (s(a , b ), Nk ) are conjugate under GLn (C) if and only if either they are
equal, or k = k = n/2 and (a , b ) = (b, a). In the latter case, solving (9.10.1) shows that the
corresponding triples both have t = 0 and z in one is replaced by z −1 in the other, whence the
triples are equivalent, which shows that f is well defined. Since f, f are clearly inverses, both
are bijections, and the result now follows from Theorem (9.8). ✷
(9.13) Remark. – There is a close analogy between the set P described here and the
“multisegments” of Zelevinsky [31, §4]. We shall elaborate on this in Section 11 below.
10. Irreducible modules and decomposition numbers
We begin by reviewing the main points concerning the representation theory of the algebra
Tna (q). In this section, R is a field.
+
(10.1) D EFINITION. – Let Λa (n) be the set
Λa (n)+ = (t, z) | t ∈ Z0 , 0 t n, n − t ∈ 2Z; z ∈ R× .
Define Λa (n) by
(10.1.1)
a
Λ (n) =
if q 2 = −1,
Λa (n)+
Λa (n)+ \ {(0, ±q)} if q 2 = −1.
Define the equivalence relation ≈ on Λa (n)+ as that which identifies (0, z) and (0, z −1 ) for all
z ∈ R× , and write
0
(10.1.2)
Λa (n) = Λa (n)/ ≈,
Λa (n)0+ = Λa (n)+ / ≈ .
Recall (e.g. from (6.6) above) that if Xt is the Tna (q)-module with basis all monic affine
diagrams : t → n, the cell module Wt,z (n) for Tna (q) is defined as the quotient of Xt by the ideal
Iχ := {γτt − χγ} for γ ∈ Xt , where χ = χz = z if t = 0 and χ = χz = z + z −1 if t = 0. There
is a bilinear map φt,z : Wt,z (n) × Wt,z−1 (n)→R which [14, (2.7)] is invariant in the sense that
φt,z (wµ, ν) = φt,z (µ, w∗ ν), (µ, ν ∈ Wt,z (n), w ∈ Tna (q)) where d∗ denotes the reflection of a
diagram d ∈ T in a horizontal line. Note that w → w∗ is an anti-automorphism of Tna (q), which
preserves TLan (q). If R is a field, the irreducible Tna (q)-modules have the following description.
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(10.2) T HEOREM [14, (2.8)]. – Let R be an algebraically closed field and maintain the above
notation. For (t1 , z1 ), (t2 , z2 ) ∈ Λa (n)+ , Wt1 ,z1 (n) ∼
= Wt2 ,z2 (n) if and only if (t1 , z1 ) ≈ (t2 , z2 ).
For (t, z) ∈ Λa (n)+ , Lt,z (n) := Wt,z (n)/ rad φt,z is either an irreducible Tna (q) module or zero,
and Lt,z (n) = 0 if and only if (t, z) ∈ Λa (n) (see (10.1.1)). All irreducible Tna (q) modules are
realised thus, and if (t1 , z1 ) ≈ (t2 , z2 ), then Lt1 ,z1 (n) ∼
= Lt2 ,z2 (n).
0
It follows that the distinct irreducible Tna (q)-modules are parametrised by Λa (n) , while the
0+
distinct cell modules are parametrised by Λa (n) . Where there is little danger of confusion,
0+
we abuse notation by denoting the elements of Λa (n) as pairs (t, z), rather than equivalence
classes of pairs. Thus we speak of Wt,z (n) and Lt,z (n) for (t, z) ∈ Λa (n)0+ . It follows from
na (q)-module Ms,N ((s, N ) ∈ P),
(e.g.) (9.12) that to understand the composition factors of the H
a
it suffices to understand those of the Tn (q)-module Wt,z (n) for appropriate (t, z) ∈ Λa (n)+ . We
therefore now turn to a description of these, which is available from [14].
◦
Let be the partial order on Λa (n)+ which is generated by the preorder ≺ which stipulates
◦
that (t, z) ≺ (s, y) if
(10.3a)
(10.3b)
0 t s n,
2
z =q
s = t + 24 (4 ∈ Z, 4 > 0) and
and y = zq −ε(s,z)
ε(s,z)s
for ε(s, z) = ±1.
Note that (10.3a,b) implies that
(10.4a)
y 2 = q ε(s,z)t
and z t = y s
and
(10.4b)
t
(t , z ) ⇒ z t = (z ) .
(t, z)
◦
It suffices to verify (10.4b) when (t, z) ≺ (t , z ), in which case it follows easily from (10.3a,b).
+
It is easily verified that [15, 4.1] the partial order on Λa (n) induces a partial order, also
denoted , on the set Λa (n)0 = Λa (n)/ ≈.
The following result is proved in [14, Theorem 5.1].
(10.5) T HEOREM. – Let R be a field of characteristic 0 or p > 0, where pe > n and e is
the multiplicative order of q 2 . Then, in the Grothendieck group Γ(Tna (q)), we have for any
+
(t, z) ∈ Λa (n) ,
(10.5.1)
Ls,y (n).
Wt,z (n) =
(s,y)∈Λa (n)0
(t,z)(s,y)
Thus the matrix expressing the cell modules in terms of the irreducibles in Γ(Tna (q)) is upper
unitriangular, and has entries 0 or 1. Now if (t, z) is confined to Λa (n), the relation (10.5.1) can
clearly be inverted.
We also have (cf. [15, Theorem (4.5)])
(10.6) T HEOREM. – In the notation above, if (t, z) ∈ Λa (n),
Lt,z (n) =
(s,y)∈Λa (n)0
(t,z)(s,y)
where ns,y
t,z = 0 or ±1.
4e SÉRIE – TOME 36 – 2003 – N◦ 4
ns,y
t,z Ws,y (n)
DIAGRAM ALGEBRAS AND DECOMPOSITION NUMBERS
517
We next explore the implications of these results for the standard modules Ms,N .
(10.7) D EFINITION. – For any triple (t, α, z) where (t, z) ∈ Λa (n)+ and α ∈ C× , define the
modules
(10.7.1)
Wt,z (α) := θα∗ Wt,z (n),
Lt,z (α) := θα∗ Lt,z (n).
(10.8) L EMMA. – All composition factors of the cell module Wt,z (α) are of the form
θα∗ Lt ,z (n) for some triple (t , α, z ).
Proof. – It is clear that the pullback via θα of a composition series for Wt,z (n) is a composition
a (q)-module Wt,z (α). The result therefore follows from Theorem (10.2). ✷
series for the H
n
We need to identify which among the Lt,z (α) are distinct. For this, we shall regard the modules
a (q)-modules, via the commutative diagram
we are considering as TL
n
(10.9)
na (q)
H
θ
θα
a (q)
TL
n
φα
incl
TLan (q)
incl
Tna (q)
a (q) defined immediately
where θ = η3 ◦ ι, η3 is as in (5.6) or (5.9), ι is the involution of H
n
preceding the statement of Theorem (9.6) above and θα is defined in (9.9).
na (q)-modules Lt,z (α) = θα∗ Lt,z (n) for (t, z) ∈ Λa (n),
(10.10) P ROPOSITION. – The H
a (q).
a (q)-modules which factor through TL
α ∈ C× , are precisely those irreducible H
n
n
a (q)-module is
Proof. – The statement is equivalent to the assertion that any irreducible TL
n
a (q)of the form φ∗α Lt,z (n). To prove this, let M be any irreducible finite dimensional TL
n
a (q) and
module. Recall (see (4.2) above) that TLan (q) = C1 , . . . , Cn is a subalgebra of TL
n
a (q) ∼
TL
= CV ⊗C TLan (q), where V is the automorphism of TLan (q) which permutes the Ci
n
cyclically.
Let M1 be an irreducible TLan (q)-submodule of M . Then by [15, (2.3)], M1 ∼
= Lt,z (n) or
V
∼
√
L
M1 ∼
(n).
If
M
(n),
then
by
[15,
(2.5)]
the
twist
M
of
M
by
the
automorphism
= L±
=
1
t,z
1
1
0, −1
V of TLan (q) is isomorphic to Lt,z (n). It follows from [15, Theorem (2.6)] that as a vector space,
M = Lt,z (n). Moreover by Schur’s Lemma, V acts as a scalar multiple of τn , i.e. as ατn for
some α ∈ C× , so that M ∼
= φ∗α Lt,z (n). In the other case, [op. cit. Theorem (2.6)] shows that
+√
∼
√
as vector space M = L0, −1 (n) ⊕ L−
(n), and following the argument in loc. cit. further
0, −1
∗
√
∼
shows that M = φα L0, −1 (n) for some α ∈ C× . ✷
0 the set of triples (t, α, z) in Ω
(see the definition
(10.11) D EFINITION. – Denote by Ω
a
and we write
in (9.12)) such that (t, z) ∈ Λ (n). Then Ω0 is a union of ∼-classes of Ω
Ω0 = Ω0 / ∼ for the corresponding set of equivalence classes.
are by Theorem (10.2) precisely those such that the corresponding
The triples (t, α, z) in Ω
module Lt,z (α) is non-zero.
We are now in a position to determine the coincidences among the irreducible modules Lt,z (α)
for different values of the parameters.
ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE
518
J.J. GRAHAM AND G.I. LEHRER
0 (see Definition (10.11)).
(10.12) T HEOREM. – Let (t, α, z) and (t , α , z ) be two triples in Ω
a (q)-modules Lt,z (α) and Lt ,z (α ) are isomorphic if and only if the triples
The irreducible TL
n
(t, α, z) and (t , α , z ) are equivalent, i.e. represent the same element of Ω (see (9.12)).
Proof. – We first show that if (t, α, z) ∼ (t , α , z ) then Lt,z (α) ∼
= Lt ,z (α ). For this, observe
∼ Wt,−z (−α) for all triples (t, α, z) ∈ Ω,
∼ W0,z−1 (α) for all α, z,
that W0,z (α) =
that Wt,z (α) =
∼ Wn,y (y −1 zα) = Ln,y (y −1 zα). It follows by taking top quotients
and that Ln,z (α) = Wn,z (α) =
that if (t, α, z) ∼ (t , α , z ) then Lt,z (α) ∼
= Lt ,z (α ).
0 , the equivalence class
To prove the converse, we need to show that for any triple (t, α, z) in Ω
∗
of (t, α, z) is determined by the isomorphism class of φα Lt,z (n) as TLan (q)-module. Recall (cf.
a (q) and that the restriction of Lt,z (α) to
diagram (10.9)) that TLan (q) is a subalgebra of TL
n
a
TLn (q) is Lt,z (n). It follows from [14, (1.9)] (or [15], or cf. the proof of (10.10) above) that the
equivalence class of the pair (t, z) in Λa (n) under the equivalence relation ≡ is determined by
Lt,z (α), where ≡ is generated by the relations
(t, z) ≡ (t, −z) if t = 0, n,
(n, z) ≡ (n, y) for any y, z ∈ C× ,
(0, z) ≡ 0, z −1 for z ∈ C× .
(10.12.1)
a (q) acts on φ∗ Lt,z (n) as ατn ∈ T a (q) does.
Further, recall that V ∈ TL
n
α
n
Suppose for the moment that t = 0, n. Then by (10.12.1), (t, z) = (t , z ) or (t , −z ). If
(t, z) = (t , z ), then by hypothesis φ∗α Lt,z (n) ∼
= φ∗α Lt,z (n), whence there is a linear map
λ : D → D, where D is the C-vector space with basis the standard affine diagrams t → n (see [14,
a (q) structures φ∗ Lt,z (n)
§2]), which commutes with the TLan (q) action, and intertwines the TL
n
α
∗
∗
and φα Lt,z (n). But in this case the restriction Lt,z (n) of φα Lt,z (n) to TLan (q) is irreducible,
whence λ must be multiplication by a scalar, say a ∈ C. Moreover we have for E ∈ Lt,z (n),
(10.12.2)
λ(V.E) = λ(ατn .E) = aατn .E = V.λ(E) = α τn .aE = aα τn .E,
whence α = α . If (t, z) = (t , −z ), then since the restrictions of Lt,z (α) and Lt,−z (α )
to TLan (q) are isomorphic irreducible representations, any linear map λ : D → D (as above)
which intertwines the Lt,z (α) and Lt,−z (α )-actions on D is of the form aU , where U is the
map defined in the proof of Theorem (2.3) of [15], and a ∈ C (for any standard diagram E,
U.E = (−1)rank(E) E). Computation of U (V.E) in two different ways as in (10.12.2) then yields
that α = −α. This completes the proof that (t, α, z) ∼ (t , α , z ) if t = 0, n.
If t = n, Wn,z (n) is one-dimensional, and hence irreducible. Moreover V acts on φ∗α Ln,z (n)
as multiplication by αz. Hence if t = n, the isomorphism class of φ∗α Lt,z (n) is determined by αz,
whence the result for this case.
Now suppose that t = 0. If z 2 = −1 (i.e. z + z −1 = 0), then the restriction to TLan (q) of
∗
φα Lt,z (n) is irreducible, and in both cases z = z and z = z −1 an argument like (10.12.2)
shows that α = α , which shows that (t, α, z) ∼ (t , α , z ).
Finally, take z 2 = −1 and q 2 = −1. In this case the restriction M to TLan (q) of φ∗α Lt,z (n)
−
is (cf. the proof of (10.10)) the direct sum L+
0,z (n) ⊕ L0,z (n). The two (irreducible) direct
summands are interchanged by τ , and V therefore acts on φ∗α Lt,z (n) via the matrix
4e SÉRIE – TOME 36 – 2003 – N◦ 4
0
ατn
ατn
.
0
DIAGRAM ALGEBRAS AND DECOMPOSITION NUMBERS
519
a (q)-module M on which
But it is shown in [15, (2.8.1)] that the isomorphism class of the TL
n
a
TLn (q) acts according to the given structure and on which V acts via the matrix
0
α2 τn
α1 τn
0
is determined by the product α1 α2 . Hence if z 2 = −1, φ∗α L0,z (n) ∼
= φ∗α L0,±z (n) if and
only if α = ±α, which is the required statement for this case. This completes the proof of
Theorem (10.12). ✷
(10.13) C OROLLARY. – In the notation of (9.12), let (s, N ) be a pair in P (recall that this
means that N = Nk is a 2-step nilpotent matrix and s = s(a, b) is a diagonal matrix such
na (q)-module Ls,N , which is either 0 or
that sN s−1 = q 2 N ). Then there is a well-defined H
a composition factor of Ms,N , with the following properties. The non-zero modules Ls,N are
pairwise non-isomorphic, and the composition factors of all the modules Ms ,N ((s , N ) ∈ P)
are among the Ls,N . The condition that Ls(a,b),Nk be zero is that
(10.13.1) q 2 = −1,
n is even, k =
n
,
2
b = −α,
a = α,
for some α ∈ C× .
Proof. – If (s, N ) corresponds to the triple (t, α, z) ∈ Ω, take Ls,N to be the top quotient
Lt,z (α) of Wt,z (α). The fact that the non-zero modules Ls,N are distinct follows from (10.12).
Moreover, by (9.12), Ms,N and Wt,z (α) have the same composition factors. But the composition
factors of Wt,z (α) are clearly the inflations via θα of those of the Tna (q)-module Wt,z (n). The
result is now clear from (10.5), which shows that the composition factors of Wt,z (n) are all top
quotients of some W t , z (n). The last assertion concerning the cases when Ls,N = 0 is obtained
simply by translating the condition in Theorem (10.2) (viz. (t, z) = (0, ±q) when q 2 = −1) into
the language of pairs using (9.9.1). ✷
Our final task is to translate the precise results (10.3a,b) et seq. concerning the composition
factors of the modules Wt,z (n) into the language of pairs (s, N ). From the argument in the proof
a (q)), for any (t, z) ∈ Λa (n)+ and
of (10.13), it is clear that in the Grothendieck group Γ(H
n
×
α∈C ,
Wt,z (α) =
(10.14)
Ls,y (α).
(s,y)∈Λa (n)0
(t,z)(s,y)
In order to describe the composition multiplicities of the standard modules Ms,N for (s, N ) ∈ P,
we therefore need to interpret the order relation ≺ in terms of pairs.
(10.15) P ROPOSITION. – Suppose that under the correspondence of (9.12), the triples
(t1 , α, z1 ), (t2 , α, z2 ) correspond to the pairs (s(a1 , b1 ), Nk1 ), (s(a2 , b2 ), Nk2 ) respectively.
◦
Then (t1 , z1 ) ≺ (t2 , z2 ) if and only if there exists 4 > 0 and ε = ±1 such that if we write
2ki = n − ti for i = 1, 2, then
k2 = k1 − 4 0,
(10.15.1)
a1 b−1
= q t1 +εt2 ,
1
(a1 , b1 )
(a2 , b2 ) =
(b1 , a1 )
ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE
if ε = 1,
if ε = −1.
520
J.J. GRAHAM AND G.I. LEHRER
Proof. – From Eqs. (9.9.1), we see that z12 = a1 b−1
1 . Hence from Eqs. (10.3), t2 = t1 + 24 for
−t1
εt2
q
=
q
for
some
ε
=
±1.
Since z2 = z1 q −ε , Eqs. (9.10.1) show that
some 4 > 0, and a1 b−1
1
(1−ε)
−(1−ε)
a2 = a1 q
and b2 = b1 q
. The relations (10.15.1) follow after a little rearrangement.
Conversely, if the relations (10.15.1) hold, Eqs. (9.9.1) and (9.10.1) may be used as above to
deduce the relations (10.3a) and (10.3b). ✷
◦
on the set P of pairs
The relation ≺ defined by Eqs. (10.15.1) generates a partial order
(s, N ) described in (9.12). This partial order may be interpreted as ordering the G-conjugacy
classes of pairs in the obvious way; viz. two classes are related if they contain related elements.
Let P0 be the set of (conjugacy classes of) pairs (s, N ) ∈ P, where s = s(a, b), N = Nk and
(s, N ) is not of the form in (10.13.1). By Theorem (10.12) P0 parametrises the composition
factors of the standard modules Ms,N , (s, N ) ∈ P. Theorems (10.5) and (10.6) may now be
applied as follows.
(10.16) T HEOREM. – Let P and P0 be the sets of semisimple-nilpotent pairs defined in the
previous paragraph, and let be the partial order on P generated by the relation (10.15.1).
na (q)-modules,
Then in the Grothendieck group of finite-dimensional H
[Ms,N ] =
(10.16.1)
[Ls ,N ] for any pair (s, N ) ∈ P,
(s ,N )∈P0
(s,N )(s ,N )
and
[Ls,N ] =
(10.16.2)
,N
nss,N
[Ms ,N ]
(s ,N )∈P
(s,N )(s ,N )
,N
where nss,N
= 0 or ±1.
We conclude by giving some applications to the structure of the standard modules.
(10.17) C OROLLARY. –
(i) The standard modules Ms,N ((s, N ) ∈ P) are multiplicity free.
(ii) If q is not a root of unity, the standard modules have at most 2 composition factors.
(iii) In all cases, Ms,N has composition length bounded by [n/2].
(iv) The maximum composition length of Ms,N is [n/2], and therefore is unbounded as
n → ∞.
These statements are easy consequences of Theorem (10.16).
11. Concluding remarks
We first wish to explain how our results generalise certain aspects of the theory in [31], which
involves the notion of “multisegments”. Without discussing the general notion in [op. cit.], let us
define these in our context as follows.
(11.1) D EFINITION. – Let R be an integral domain and q ∈ R× . A segment in R is a sequence
(11.1.1)
I(a, m) = a, aq −2 , aq −4 , . . . , aq −2(m−1)
(a ∈ R× , m ∈ Z>0 ) of elements of R× . In addition, there is a unique segment of length 0, which
is the empty sequence. In what follows, some remarks apply only to segments of length = 0, and
4e SÉRIE – TOME 36 – 2003 – N◦ 4
DIAGRAM ALGEBRAS AND DECOMPOSITION NUMBERS
521
we shall rely on the context to distinguish these. Note that the elements of I(a, m) are distinct if
q is not a root of unity. Otherwise, there may be repetitions in I(a, m). Denote by |I(a, m)|
the multiset (i.e. set with multiplicities) underlying I(a, m). Then I(a, m) is determined by
|I(a, m)|, together with its initial element.
The length of I(a, m) is m, and we may speak of a subsegment of I(a, m), with the obvious
meaning, and similarly for initial and final subsegments of I(a, m).
Suppose I(a1 , m1 ) and I(a2 , m2 ) are two segments. We say that I(a1 , m1 ) precedes
I(a2 , m2 ) if a2 = a1 q −2i for some i ∈ {m1 − m2 + 1, m1 − m2 + 2, . . . , m1 }, where only the
non-negative among these integers are included; i.e. if the initial element of I(a2 , m2 ) coincides
with one of the final m2 elements of I(a1 , m1 + 1). If I(a1 , m1 ) precedes I(a2 , m2 ), we may
define a concatenation I(a1 , m1 )#I(a2 , m2 ) as any segment which starts with I(a1 , m1 + 1)
and finishes with I(a2 , m2 ). Such a segment exists by definition, and any concatenation clearly
has length strictly greater than the maximum of m1 and m2 .
Say that two segments I(a1 , m1 ) and I(a2 , m2 ) are linked if either one precedes the other.
Observe that when q is a root of unity, each may precede the other, and there may be many
different concatenations I(a1 , m1 )#I(a2 , m2 ) and I(a2 , m2 )#I(a1 , m1 )
Note that in the “generic” case, when q is not a root of unity, this definition (of linked segments)
coincides with the definition [31, 4.1, p. 184].
We may now state our results concerning composition factors as follows. First note that
the pairs (s(a, b), Nk ) ∈ P (cf. (9.12)) correspond bijectively to pairs I1 = I(a, n − k) and
I2 = I(b, k) of segments of length n − k = t + k and k respectively, where n − k = t + k k.
In the statement below, we write MI1 ,I2 and LI1 ,I2 respectively for the corresponding standard
modules Ms(a,b),Nk and irreducibles Ls(a,b),Nk (see Section 10 above).
(11.2) T HEOREM. –
(i) The standard module Ms,N is irreducible if and only if the segments I1 , I2 are not linked.
(ii) If I1 precedes I2 , let I1 = I1 #I2 be any concatenation of I1 and I2 . This is a segment of
na (q)-module
length n− k , where k < k (see above). Define I2 = I(b, k ). Then the irreducible H
LI1 ,I2 is a composition factor of MI1 ,I2 .
(iii) If I2 precedes I1 , take I1 to be any concatenation I2 #I1 . This is a segment of length
a (q)-module LI ,I is
n − k , where again k < k. Define I2 = I(a, k ). Then the irreducible H
n
1 2
a composition factor of MI1 ,I2 .
(iv) All composition factors of MI1 ,I2 have multiplicity one, and are of the form LI1 ,I2 , where
(I1 , I2 ) arises from (I1 , I2 ) by a sequence of transformations of the form described in (ii) and (iii)
above.
(v) We have LI1 ,I2 = 0 if and only if q 2 = −1 and I2 = −I1 , so that n = 2k.
(vi) Say that (I1 , I2 )
(I1 , I2 ) if (I1 , I2 ) is obtained from (I1 , I2 ) by a sequence of
transformations of the form (ii) and (iii) above. Then in the Grothendieck group of finite
na (q) modules, we have
dimensional H
[LI1 ,I2 ] =
I ,I nI11 ,I22 [MI1 ,I2 ]
(I1 ,I2 )(I1 ,I2 )
I ,I where nI11 ,I22 = 0 or ±1.
These statements are simply restatements of the facts treated in (10.15) and (10.16) above.
Note the similarity between our Theorem (11.2)(i) and [31, Theorem 4.2]. The latter is of
course false when q is a root of unity without our modification of the meaning of linked segments.
Note also that our statement holds for any q ∈ C, even q = 1.
ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE
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J.J. GRAHAM AND G.I. LEHRER
(11.3) We shall now give a more explicit description of the poset P of bisegments P = (I1 , I2 ),
ordered by the relation , with a view to giving a more explicit version of Theorem (11.2)(vi). It
is convenient to change notation as follows. We have defined a segment above as
where f = aq −2(m−1) .
I(a, m) = a, aq −2 , . . . , f
Write [a, f ]m for this segment. This notation applies when a, f ∈ R× and m is a positive integer
such that f = aq −2(m−1) . We call a and f the endpoints of the segment. The set P of bisegments
is parametrised by unordered pairs P = {[a, f ]l , [b, q]k } of segments such that k + l = n. We say
P has length n = k + l and write k(P ) =min(k, l). The corresponding standard module and
irreducible modules will be denoted MP , LP respectively.
Suppose P and Q are bisegments of the same length. We say that P and Q are transverse
if they are of the form {[a, f ]l , [b, g]k } and {[a, g]i , [b, f ]j } respectively; we say P and Q are
parallel if they are of the form {[a, f ]l , [b, g]k } and {[a, f ]i , [b, g]j } respectively. (Note that the
subscripts may be zero.)
Given a pair Q, Q of parallel bisegments of the same length, we say that Q and Q have the
same orientation if they are of the form {[a, f ]l , [b, g]k } and {[a, f ]i , [b, g]j } respectively with
k l and j i; otherwise Q and Q have opposite orientations.
The following assertions concerning the combinatorics of the poset P may be easily checked.
(i) If P, P ∈ P and P P , then P is either transverse or parallel to P (possibly both).
◦
(ii) If P is transverse to P and k(P ) < k(P ), then P ≺ P .
(iii) If P is transverse to itself, then {P ∈ P | P P } is totally ordered. If P1 is the minimal
element of {P ∈ P | P ≺ P } (i.e. P1 = P ) then
[LP ] = [MP ] − [MP1 ].
(iv) If P, P are parallel, have the same orientation, k(P ) < k(P ), and there exists Q with
◦
◦
P ≺ Q, then there is a bisegment P1 ∈ P such that P ≺ P1 ≺ P .
(v) Say that P = P ∈ P are opposed if they are parallel and not comparable. Then given
Q ∈ P such that there exists P ∈ P with Q ≺ P , there exists at most one element Q ∈ P such
that Q, Q are opposed.
The proof of (v) proceeds by showing that if Q is opposed to both Q and Q then Q , Q
are opposed to each other, and hence by (iv) they must have opposite orientations. But they both
have orientation opposite to that of Q, whence two such elements Q , Q cannot exist.
(vi) Suppose that P1 is not transverse to itself. If P1 ≺ Q ≺ P2 , then there exists a unique
bisegment Q which is opposed to Q.
(vii) If P ≺ Q, then the number of bisegments R in the interval [P, Q] := {R | P R Q}
which are parallel to P is equal to the number of R which are transverse to P .
The structure of the poset P is completely determined by the properties (i) to (vii) above,
which are all straightforward to prove. Moreover it is an easy consequence of (vii) that
(11.4) P ROPOSITION. – Suppose P ∈ P is not transverse to itself. Then
Q
nP [MQ ],
[LP ] =
P Q
where
nQ
P =
4e SÉRIE – TOME 36 – 2003 – N◦ 4
1
if Q is parallel to P ,
−1 if Q is transverse to P .
DIAGRAM ALGEBRAS AND DECOMPOSITION NUMBERS
523
We remark also that given the results of [20,1,10], our statement (11.2)(vi) implies that the
Kazhdan–Lusztig polynomials which arise from intersection complexes on the closure of the
ZG (s) orbit of the nilpotent element N are equal to 1.
Finally, we observe that the results of this paper may be used to discuss aspects of the
na (q) over any algebraically closed field of
representation theory of the affine Hecke algebra H
positive characteristic, i.e. the “modular case”. This is carried out for the algebras TLB n (q, Q)
in [6].
Acknowledgement
Both authors thank the Australian Research Council for support. The second author thanks the
Alexander von Humboldt Stiftung for support and the Universität Bielefeld for hospitality during
the preparation of this work.
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(Manuscrit reçu le 12 décembre 2001 ;
accepté, après révision, le 3 juillet 2002.)
J.J. G RAHAM
School of Mathematics and Statistics,
The University of Birmingham,
Birmingham B15 2TT, UK
G.I. L EHRER
School of Mathematics and Statistics,
University of Sydney,
Sydney, N.S.W. 2006, Australia
E-mail: [email protected]
4e SÉRIE – TOME 36 – 2003 – N◦ 4