Download AFFINE LIE ALGEBRAS, THE SYMMETRIC GROUPS, AND

AFFINE LIE ALGEBRAS, THE SYMMETRIC GROUPS,
AND ALGEBRAIC GROUPS
MALKA SCHAPS
1. Ordinary Representations of Groups
For any finite groups G = {g1 , . . . , gs } and field F , a representation
of G over F is a group homomorphism
ρ : G → GLr (F ),
and r is called its degree. Let F G be the corresponding group algebra, with F -vector space basis hg1 , . . . , gr i and algebra multiplication
induced by the group multiplication law and linearity. Then the representation ρ induces an algebra homomorphism
ρ̃ : F G → GLr (F ),
P
P
where
ai gi ∈ F G is mapped to
ai ρ(gi ). The representation ρ can
be recovered from ρ̃ by restricting to G, and we will consider them
interchangeable. The most compact description of ρ or ρ̃ is given by
specifying the images of a set of generators of G, as matrices satisfying
the defining relations of the group.
Two representations ρ and τ are considered equivalent if they differ
only by a change of basis in the underlying vector space F r on which
the matrices act. More formally, ρ τ if there exists an invertible r × r
matrix P such that for each G ∈ G, tau(g) = P −1 ρ(g)P . The “atomic
particles” of representation theory are the indecomposable representations, those which are not equivalent to a direct sum of representations.
We assume, for the remainder of this section, that F satisfies the
following two conditions, in which case we say that we are studying
the ordinary representations of the finite group G.
• (generic characteristic): F is a field of characteristic 0, or of
positive characteristic p such that p - |G|.
• (F sufficiently large): F contains a primitive root of unity ζ for
|G| = s.
If these two conditions hold, then the indecomposable representations are, in fact, irreducible, having no subrepresentations, and then
by Maschke’s theorem, we can write F G as a direct sum of simple
Date: Nov. 16-30, ’10.
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algebras and decompose F G in the form
FG ∼
=
t
M
Mni (F ).
i=1
Each component determines a unique irreducible representation ρi and
these give a complete set of representatives of irreducible representations. The size ni of the matrix block is called the degree of the representation. The inverse images ei of the identity matrices Ini of the
t components form a basis for the center Z(F G) of the group algebra
F G.
A representation ρ is completely determined by its character χρ , a
map from G into F given by
χρ (g) = tr(ρ(g)).
Since the trace is invariant on conjugacy classes of matrices, χρ is a class
function on G, that is, fixed on conjugacy classes. The character of the
irreducible representation ρi will be denoted by χi for brevity. The
addition of characters corresponds to direct sums of representations,
and the componentwise product of characters corresponds to tensor
products of representations.
The conjugacy class sums form an alternative basis for the center
Z(F G), and thus there are exactly t conjugacy classes. We will order
the elements of G so that g1 is the identity element e of the group and
gj , for j = 1, . . . , t are representatives of the distinct conjugacy P
classes.
The conjugacy class [gj ] will be denoted by Cj and its class sum g gj g
bj .
will be denoted by C
We recall that we have assumed that F is of generic characteristic
and is sufficiently large. The character table X is the t × t square
matrix [χi (gj )]. For any i, j the character value χi (gj ) is the sum of the
eigenvalues of ρi (gj ), and since gj is of finite order dividing the order
s of G, the entries in the character table can be expressed as powers
of ζ, making them more or less independent of the actual choice of the
field F , as long as it satisfies our assumptions. If N is the diagonal
matrix with the degrees along the diagonal, and M is the diagonal
matrix with conjugacy class sizes mj = |Cj | along the diagonal, then
W = N −1 XM , the matrix of central characters, gives a base change
matrix between the two bases of Z(F G), in the sense that
X
Ĉj =
wij ei .
Since W, N and M are all invertible, so is X.
2. Ordinary representations of the symmetric groups
For the symmetric groups Sd , the conjugacy classes consist of all
elements with the same cycle structure. Each cycle structure can be
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represented by a partition of d, and thus there is a natural one-toone correspondence between conjugacy classes and partitions. It is
considerably more difficult, though standard, to show that the irreducible representations are also in one-to-one correspondence with the
partitions of d, and we will assume this to be known. The character
corresponding to a partition λ will the denoted by χλ , and the corresponding degree by nλ . The conjugacy class class representative with
the integers {1, . . . , n} inserted in cycles of lengths λ1 , . . . , λk will be
denoted by gλ . The representation corresponding to the partition with
one part, (d), is the trivial representation of degree 1, sending every
group element to 1, and the representation corresponding to the partition with d parts, (1, 1, . . . , 1) is the alternating representation, also
of degree 1, which sends each group element to 1 if the permutation is
even and to −1 if the permutation is odd.
The partition is represented by its Young diagram. The partition is
regarded as having been built up one node at a time from the empty
partition, in such a way that each intermediate diagram is a partition.
Each such path is recorded by placing the integers from 1 to n in the
nodes of the diagram, creating a standard tableau in which the integer
are increasing on every row and column. The degree nλ of the representation ρλ is the number of such paths and the set of standard tableaux
may be taken as a basis. The defining property of this representation is that it is simultaneously induced from the trivial representation
of the subgroup of Sd stabilizing the rows and from the alternating
representation of the subgroup of Sd stabilizing the columns.
We now leave the standard material which has been known in essence
since the time of Frobenius 150 years ago, and describe a new approach
to the representations of the symmetric group which goes back about 15
years. We retain the Young diagrams, but add a new label, a residue to
each node. These new labels will be constant on diagonals, will increase
by 1 to the right and will decrease by 1 as we go down. For simplicity,
we will give the main diagonal the residue 0. Each Young diagram is
uniquely described by the lengths of its diagonals, which are given by
its content, a list of the number of nodes of each residue. A standard
tableau is described by a list of the residues which are encountered as
one builds up the Young diagram node by node.
We now use this new labeling of nodes to connect the representations
of the symmetric groups to the weights of an infinite dimensional Lie
algebra A∞ . This Lie algebra is formed like the Kac-Moody algebras we
have been studying, except that it has an infinite Cartan matrix, which
looks like the classic Cartan matrix of type A extended infinitely in
both directions, and an infinite Chevalley basis consisting of the union
of sets {ei , fi , hi )} for i ∈ Z. There is also an infinite set of elements
. . . , Λ−2 , Λ−1 , Λ0 , Λ1 , Λ2 , . . . ,
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MALKA SCHAPS
which are dual to the hi as functionals, in the sense that hhi , Λj i = δij .
We consider a highest weight module
P for this Lie algebra, with highest weight Λ0 . The weight µ = Λ0 − i∈Z γi αi will be a weight of a homogeneous element of this module if and only if γ = (. . . , γ−1 , γ0 , γ1 , . . . )
is a sequence of positive integers with only a finite number of nonzero
terms, and if, furthermore, it is the content of a partition of some nonnegative integer d. The element v ∅ is the unique basis element of weight
Λ0 and is referred to by physicists as the vacuum vector. The basis elements with weight µ will be parameterized by sequences i which will
determine the standard tableaux. The same sequence i determines the
representation of the basis element
v i as a product
fid fid−1 . . . f1 v ∅ .
The number of sequences producing the weight µ is exactly the degree
of the corresponding representation of Sd . At any partition λ0 along the
way, corresponding to a weight µ0 with content γ 0 , the addable nodes
are those with residue i for which hµ0 , hi i > 0. The only possible values
are −1, 0, 1 and the sum of all the values must be 1, so it is always
possible to continue.
Example 2.1. We give the residues for the partition (4, 2, 1). For
compactness we use a, b, . . . to designate −1, −2, . . . .
0 1 2 3
a 0
b
The standard tableau
1 2 3 4
5 6
7
corresponds to a sequence i = (0, 1, 2, 3, −1, 0, −2).
The standard tableau
1 4 6 7
2 5
3
corresponds to a sequence i = (0, −1, −2, 1, 0, 2, 3).
3. Modular representations of the symmetric groups and
Iwahori-Hecke algebras at root of unity
We now turn to the representation theory of the symmetric group
over a field whose characteristic e does divide d!, the so-called modular
case. Dipper and James discovered that in this case the representation
theory is similar to that of an Iwahori-Hecke algebra in which ξ is an
e-th root of unity in F (which has characteristic different from e). The
symmetric group case is called the degenerate case, and the remaining
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values of ξ give non-degenerate Hecke algebras, but the two cases can
be treated together with only minor modifications.
We now define the Iwahori-Hecke algebra Hd (F, ξ). Let ξ be an
element of F ∗ . We define an algebra Hd with generators T1 , . . . , Tn−1
over the field F , using relations
(Ti + 1)(Ti − ξ) = 0, 1 ≤ i ≤ n − 1
Ti+1 Ti Ti+1 = Ti Ti+1 Ti 1 ≤ i ≤ n − 2
Ti Tj = Tj Ti , 1 ≤ i < j ≤ n − 1, |i − j| ≥ 2
If ξ is not a root of unity, then the representation theory of Hd
is essentially the same as the ordinary representation theory of Sd ,
with the algebra being semi-simple and the irreducible representations
parameterized by the partitions of d. If the field F is generic for Sn
and ξ = 1, then we get the symmetric group. There is an analog of this
construction for the so-called cyclotomic Hecke algebras HdΛ , in which
the blocks correspond to weights with highest weight Λ but we will not
go into that theory in these lectures.
The main difference is in the meaning of the residues label the nodes
of a partition. These are now taken to be elements of I = Z/eZ,
which we will represent by numbers in the set {0, 1, . . . , e − 1}. In the
degenerate case they represent elements of the base field of F , and in
the nondegenerate case they represent powers of ξ. More formally, we
define
(
i (if ξ = 1)
ν(i) ≡
ξ i (if ξ 6= 1).
These elements ν(i) enter the theory as the eigenvalues of the following elements of Hd , known as Jucys-Murphy elements:
(
Lr ≡
(1, r) + (2, r) + · · · + (r − 1, r)
ξ 1−r Tr Tr−1 . . . T2 T1 T1 T2 . . . Tr−1 Tr
(if ξ = 1)
1 ≤ r ≤ d.
(if ξ =
6 1);
Note that in the degenerate case, L1 = 0 and L2 = (1, 2). The
Lr commute with each other and form a commutative subalgebra Z
of Hd known as the Gelfand-Zetlin subalgebra. The restriction of any
Hd module M to this subalgebra breaks up into a direct sum of linear representations, with eigenvalues of the form ν(i). The module
induced from an irreducible module for Hd−1 will be a direct sum of
indecomposable projective modules for Hd , each with irreducible socle
and each socle with a single eigenvalue of the Jucys-Murphy element
Ld . It is this choice of idempotents which distinguishes among various
components in the induced modules.
As before, the blocks correspond to weights in an affine Lie algebra,
(1)
but now the algebra is Ae−1 . This has a Dynkin diagram which is a
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MALKA SCHAPS
cycle of length e. The Cartan matrix is the Cartan matrix of Ae with
an additional −1 at each end of the off diagonal. Thus for e = 5 it
would be


2 −1 0
0 −1
 −1 2 −1 0
0


 0 −1 2 −1 0 


 0
0 −1 2 −1
−1 0
0 −1 2
In the modular case, it is no longer true that every indecomposable
representation is irreducible. The blocks now correspond to collections
of partitions with a common core, obtained by removing a fixed number
w of rim hooks of length e. Each rim hook will contain a complete set
of residues. In terms of the affine Lie algebra, removing rim hooks
corresponds to adding copies of the null root.
The irreducible modules correspond to those partitions which are erestricted, i.e., do not have e consecutive columns of the same length.
(This is dual to the more traditional notion of e-regular, under the
duality which flips the partitions across the main diagonal.) There is
a diagram called the crystal graph, due originally to Kashiwara, which
describes the way that the irreducible representations are built up by
induction.
Example 3.1. Let us consider the Young tableau in Example 2.1 for
e = 3.
0 1 2 0
2 0
1
In this case there are three simple roots α0 , α1 , α2 to the affine Lie
algebra, and the null root is δ = α0 + α1 + α2 .The corresponding weight
in the affine Lie algebra is
Λ0 − 3α0 − 2α1 − 2α2 .
We can remove two rim hooks, which corresponds to adding two copies
of δ, giving Λ0 − α0 .
−−−−
2 0
1
,− 1 2 0, 0.
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We will now give that part of the crystal graph containing all the paths
leading to (4, 2, 1).
∅
↓
0
.
&
0
2
0 1
&
.
0 1 2
2
↓
.
0
2 0 1
1, 2
↓
0 1
2
1
0 1
2 0
.&
.↓
0 1 2
2
1
0 1 2
2 0
↓
&
&
↓
0 1
2 0
1
.
0 1 2
2 0
1
0 1 2 0
2 0
&
↓
0 1 2 0
2 0
1
Each i-string, for i ∈ I, is actually an sl2 -representation. As one
goes down an i-string, the i-nodes are added starting at the bottom
left and working along to the upper right. We give a few strings of this
type. Note that the second 0-string is not contained in the diagram
because it does not lead to the partition (4, 2, 1).
0
0 1
0
2
2
i=1: 2 → 1 → 1 .
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MALKA SCHAPS
0 1
0 1 2
0
1
2
i=2:
→
→ 2
.
0 1
0 1
2
i=0:
→ 2 0.
0
0
1
1
2
2
→ 0.
i=0:
We recall that the finite Lie algebra corresponding to an affine Lie
algebra is obtained by removing the simple root α0 , the corresponding Chevalley generators, and the first row and column of the Cartan
matrix. In terms of the crystal graph, we define a floor to be the
set of all vertices with a fixed integral coefficient for α0 . This will
then correspond to a homogeneous basis for a representation of the
finite Lie algebra, which in our case is sle . The irreducible representations of sle are highest weight representations, with highest weight
λ1 Λ1 + · · · + λe−1 Λe−1 , for some partition λ of an integer d. The length
of an i-string going out from the highest weight vector will be λi . Most
of the floors will correspond to reducible representations of sle , but even
these will be highly symmetrical because the finite Weyl group will act
on each floor. For sl3 , the irreducible representations can be diagramed
as irregular hexagons with alternating sides of integral lengths a and
b. The finite Weyl group is S3 , where the axes of the reflections are
the lines connecting the midpoints of opposite sides, and the 3-cycles
permute the sides of equal length. If b = 0, the hexagon reduces to a
triangle. One can find such a hexagon in the diagram above by considering all partitions with 0-content equal to 1.
There is also a reduced version of the crystal graph in which the
vertices correspond to blocks of the symmetric groups instead of to irreducible representations. Let P (Λ) be the set of weights of occurring
in a highest weight module with highest weight Λ. The reduced crystal graph has a geometric representation in which the vertices are the
integer lattice points in Re determined by the contents in P (Λ).
Example 3.2. If we redo the diagram in Example 3.1 as a part of
the reduced crystal graph, letting a block with core ν and weight w be
represented by ν w , we get the following, where we have now included all
blocks for d ≤ 7, including one which was not in the previous diagram.
Note the symmetry between left and right, which is just taking the
dual partition and multiplying the residues by −1.
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∅0
↓
0
0
.
0
2
&
0
&
0 1
0
0 1
2
1
0
.
1
.
0 1 2
2
0
0
↓
0
2
∅
↓
&
1
.&
0 1 2
2
1
1
↓
0 1 2 0
2 0
.& ↓
&
0
&
↓
.
1
↓
0 1
2 0
0
2
∅2
↓
0
0 1
0
0
.
2
In this representation, one can see the effect of the shift by δ, which
simply adds one to each exponent. One can also see that the exponents
are symmetric along the strings, increasing toward the center.
The weight space of an affine Lie algebra has two different bases.
One is
{Λ0 , α0 , . . . , αe−1 }
and we have been representing our weights in this basis, using the
components of the content as the coordinates of the αi . The second
basis is obtained by taking the dual basis of the hi (or αi∨ ) and adding
δ, giving
{Λ0 , Λ1 , . . . , Λe−1 , δ}.
The hub of a block is the set of coordinates of the weights of the affine
Lie algebra with respect to the Λi , moding out by δ. The components
in the hub represent the weights of the hi in the sl2 representation
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MALKA SCHAPS
along the i-string. We recall that an affine lie algebra has a suitably
normalized central element c. The level of a weight µ is hµ, ci and
in our case is just the sum of the coordinates of the hub. Since the
level of any simple root is always zero, and thus in any highest weight
representation, the level is fixed. In the symmetric group case, since
the highest weight is Λ0 , which has level 1, the sum of the coordinates
of any hub is 1. The i-th coordinate of the hub corresponding to any
weight or content, gives the position of the corresponding vertex of
the reduced crystal graph along its i string, where each application of
ei increases the weight by 2 and each application of fi decreases the
weight by 2. The set of all weights for which adding a copy of δ is no
longer a weight in P (Λ0 ) is called the set of maximal weights, max(Λ0 ).
In the chart below, the exponent on each hub is the number of copies of
δ which can be added, and the elements of max(Λ0 ) are those vertices
with exponent 0.
(1, 0, 0)0
↓
(−1, 1, 1)0
.
&
(0, 2, −1)0
(0, −1, 2)0
&
.
(1, 0, 0)1
. ↓
0
&
1
(2, 1, −2)
(−1, 1, 1)
↓
.&
(0, 2, −1)1
↓
&
(−2, 3, 0)0
(2, −2, 1)0
.& ↓
(3, −1, −1)0
↓
.
(1, 0, 0)2
& ↓
(0, −1, 2)1
↓
(−2, 0, 3)0
.
(−1, 1, 1)2
4. Representations of algebraic groups
We now come to the algebraic groups. We give a description in
general of algebraic groups, before restricting to the finite groups among
them. An algebraic group over a field L is a subgroup of GL(n, L) which
can be described by polynomial equations {r1 (x), . . . , ru (x))}, with x =
xi j ni,j=1 . Associated to the group is a coordinate ring, a commutative
ring A = L[x, y]/(r1 , . . . , ru , det(x)y − 1). The most commonly studied
groups are the general linear groups, orthogonal or unitary groups and
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symplectic or spin groups, each category having its special (determinant
1) and projective (no scalars) versions.
The field L can be R, C, a p-adic field, or a finite field Fq , for some
prime power q, in which case the group is finite. The groups over R are
called Lie groups and are studied with the tools of differential geometry.
The groups over C are called complex Lie groups and are studied with
the tools of algebraic geometry. Those over p-adic fields are studied
with the tools of algebraic number theory, while the finite groups are
generally the province of group theorists and combinatorists.
The representation theory is simplest for the simple complex Lie
groups SLn (C), SOn (C), and SP2 n(C), the so-called classical groups.
In this case the representations are controlled by the corresponding
Lie algebras sln (C), son (C), and sp2n (C). These representations are
parameterized by partitions and are intimately connected to the representations of the symmetric group parameterized by the same partition
in a way that we will now explain
4.1. The Weyl construction. The complex Lie group in GLn (C)
has a natural representation V of dimension n, which is given by matrix multiplication on Cn on the left, where the vectors are written a
columns. On the other hand, the tensor product V ⊗d has a natural
action of Sd on the right by permutation of factors. V ⊗d Let F be any
field generic for Sd , which surely includes C. We know that
M
F Sd ∼
Uλ ,
=
where uλ ∼
= Mnλ (F ), and where nλ is the degree of the representation
ρλ , which is given by the number of standard tableaux. As a left
F Sd -module, Uλ is a direct sum of nλ copies of the irreducible module
corresponding to the representation ρλ .
For any vector space V over F , we have
(1)
V ⊗d = V ⊗d ⊗F Sd F Sd
(2)
= V ⊗d ⊗F Sd ⊕Uλ
(3)
= ⊕V ⊗d ⊗F Sd Uλ
Each of the components of this direct sum is a GL(V ) modules, and
decomposes into a direct sum of nλ isomorphic GL(V ) − Sd modules.
In order to choose just one of these, we must go into more detail about
the representation of Sd corresponding to a given partition λ, and, in
fact, to a specific standard tableau T for λ.
Let us define PT to be the set of elements of Sd which preserve the
rows of T when operating on the numbers in the nodes of the Young
diagram of λ defining T , and let Qλ be the subgroup of Sd preserving
the columns of T . In the first tableau for (4, 2, 1) in Example 3.1, we
would have
PT = S{1,2,3,4} × S{4,5} × S{7}
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MALKA SCHAPS
and
QT = S{1,5,7} × S{2,6} × S{3} × S{4} .
We now define two elements of Sd associated to the tableau T ,
X
aT =
σ,
σ∈PT
and
bT =
X
sgn(σ)σ.
σ∈QT
The product
cT = a T bT
is the Young symmetrizer of the tableau. It is a scalar multiple of
an idempotent of F Sd , so that for some integer nλ depending only
on the partition λ, cT cT = dλ cT . The idempotent dλ cT is, in fact,
a primitive idempotent, a fact that can be demonstrated by showing
that dimF cT F Sd cT = 1. Now, as desired, multiplying Uλ by one of
the symmetrizers cT for λ will choose one F Sd -irreducible factor from
Uλ . However, in fact, multiplying by cT will also isolate an irreducible
GL(V ) module.
We now define the Schur functor
ST (V ) = V ⊗d ⊗F Sd F Sd cT .
When F = C and V is the natural representation of GLn (C), this
will in fact isolate an irreducible representation. All the ST for T a
standard tableau for λ will give isomorphic representations, so in fact
the notation Sλ (V ) is usually used. A good exposition of this material
may be found in [2].
The representations of GL(C) obtained in this way are almost all
the irreducible representations, the remaining representation being obtained by tensoring with powers of the determinant. If d < n, then
they are all nonzero. They restrict to representations of SLn (C), and
they correspond to irreducible representations of the simple Lie algebra sln (C). In general, the representations of a simply connected Lie
group are obtained by integration from an integrable representation of
the Lie algebra at the identity, and then for a non-simply connected
Lie group, the representations are those of the simply connected group
which are invariant under the action of the group by which one takes
the quotient.
We now return to the case of finite algebraic groups, when F has
characteristic 0. One of the principal methods for forming irreducible
characters is called Harish-Chandra induction, and this was generalized to Deligne-Lusztig induction. In Harish-Chandra induction, the
representations of a Levi subgroup invariant under the action of the
Frobenius map (raising every element of the matrix to the power q) are
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extended to the corresponding parabolic subgroup, and from there induced up to the group. The induced representations are parametrized
by Levi subgroups L and by cuspidal representations of L, that is, irreducible representations of L which are not induced from any Levi
subgroup of L. The unipotent characters of GLn (Fq )are those associated to partitions of n in a manner similar to that we have seen in the
non-finite case.
Let q be a prime power, let n ≥ 0 be an and consider the group
Gn = GL(n, Fq ), where q is a power of a prime p. We assume that F
has characteristic `, where ` 6= q(q − 1). As with the modular theory of
representations of the symmetric groups, the group algebra F Gn has
a decomposition into blocks. To each block is associated a number of
irreducible characters from the characteristic 0 case. A block which
contains a unipotent character is called a unipotent block.
There is a one-to-one correspondence between unipotent blocks of
F GLn (Fq ) and blocks of Hn (F, q) [1]. The blocks correspond to the
ee−1 ,
weights in a highest weight space for Λ0 for the affine Lie algebra A
where e is the order of q in F . Chuang and Rouquier proved that the
symmetrically-placed blocks along an i-string have equivalent derived
categories [3].
References
[1] [FB] P.Fong and B. Srinivasan, The blocks of finite general linear and
unitary groups, Inventiones mathematicae, 69 (1982), 109-153.
[2] [FH] W. Fulton and J. Harris, Representation Theory, Springer-Verlag,
(1991).
[3] [CR] J. Chuang and R. Rouquier, Derived equivalences for symmetric
groups and sl2 -categorification, Annals of Mathermatics, 167 (2008), 245298.