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MATHEMATICS OF COMPUTATION
Volume 67, Number 222, April 1998, Pages 751–758
S 0025-5718(98)00919-3
THE NUMBER OF LATTICE POINTS IN ALCOVES AND
THE EXPONENTS OF THE FINITE WEYL GROUPS
RUEDI SUTER
Abstract. We count lattice points in certain rational simplices associated
with an irreducible finite Weyl group W and observe that these numbers are
linked to the exponents of W .
1. Introduction
The main point of this note is to present an (as far as I know) new formula for
the exponents of an irreducible finite Weyl group. For basic terminology the reader
is referred to any one of the books [1, 3, 5]. In any case, let me recall a few ways of
how the exponents occur. Thus let W be an irreducible finite Coxeter group with
set of Coxeter generators S and rankQl = |S|.
s ∈ W is well-defined up to conjugacy, and
The Coxeter transformation c :=
s∈S
its order h = |c| is the Coxeter number of W . If exp 2πim1 /h, . . . , exp 2πiml /h with
0 < m1 6 · · · 6 ml < h are the eigenvalues of c in the reflection representation,
then the integers m1 , . . . , ml are called the exponents of W .
Here is another approach to describe the exponents. Let ` : W → Z>0 be the
length function. Then
(1)
X
w∈W
t`(w) =
l
Y
1 − tmi +1
.
1−t
i=1
A third characterization of the exponents comes from the fact that the numbers
m1 + 1, . . . , ml + 1 are the degrees of the fundamental polynomial invariants of W .
Fourth, the exponents of W can be read off the Poincaré polynomial of the
hyperplane arrangement associated with W .
Suppose from now on that W is crystallographic. It is thus the Weyl group of a
compact, connected Lie group G. The manifold G has the same cohomology as a
product of odd-dimensional spheres,
Q
l
S 2mi +1 , R ,
H ∗ (G, R) ∼
= H∗
i=1
which again determines the exponents.
Still another possibility to describe the exponents in the crystallographic case is
as follows. Consider a root system with Weyl group W , and let ki be the number of
Received by the editor September 3, 1996.
1991 Mathematics Subject Classification. Primary 20F55; Secondary 05A15, 11P21, 11P83,
17B20, 17B67, 52B20.
Supported by the Swiss National Science Foundation.
c
1998
American Mathematical Society
751
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752
RUEDI SUTER
positive roots of height i. (The height of a root is the sum of its coefficients when
expressed as a linear combination of simple roots.) Then (ml , . . . , m1 ) is the dual
partition to the partition (k1 , . . . , kh−1 ).
2. Another formula for the exponents
Now we come to another formula for the exponents of an irreducible finite Weyl
group W , a formula which involves counting lattice points in a rational simplex
∆(m) . As an aside let me mention that in the past few years considerable progress
has been made in the theory of toric varieties and hence in the problem of counting
lattice points in integral polyhedra (see [2, 4]). The counting problem we shall deal
with is very simple.
Let me explain the definition of ∆(m) , which also provides some motivation. A
reader who is not familiar with semisimple Lie theory may hop to the beginning
of the next paragraph. Let g be a simple complex Lie algebra whose Weyl group
is W . (It doesn’t matter whether we take g = so2l+1 (C) or g = sp2l (C) if W is
hyperoctahedral.) We choose a Cartan subalgebra h and a Borel subalgebra b with
h ⊆ b ⊆ g and thus get an irreducible finite root system Φ ⊆ h∗ together with
a basis of simple roots (αi )i=1,...,l and its corresponding set of dominant integral
l
L
weights P+ =
Z>0 $i , where ($i )i=1,...,l are the fundamental weights, i. e., the
i=1
basis of h∗ dual to the basis (α∨
i )i=1,...,l of simple coroots. The intersection Φ ∩ P+
consists of the highest root θ together with the highest short root θs . Of course,
θ = θs if g is simply laced. We want to count the elements of
∆(m) := λ ∈ P+ hλ, θs∨ i 6 m
(m ∈ Z).
The coroot θs∨ is the highest root of the dual root system, and its coefficients with
∨
respect to the basis (α∨
i )i=1,...,l , i. e., the numbers ni := h$i , θs i (i = 1, . . . , l) are
sometimes called the marks (of the dual root system). Equivalently, |∆(m) | is the
number of dominant integral weights (of fixed null depth) at level m of the (twisted
if g is non-simply-laced) affine Kac-Moody algebra corresponding to g. (For the
untwisted version one has to replace θs by θ in the definition of ∆(m) . So instead of
having n1 , . . . , nl being the marks, one would take them to be the comarks.)
The upshot of the previous paragraph is that we have l positive integers n1 , . . . , nl
associated to W . The cardinality |∆(m) | we are interested in is nothing else than
the dimension of the K-algebra (K is any field)
K x1 , . . . , xl ]/(monomials of degree strictly bigger than m)
with deg xi = ni (i = 1, . . . , l).
The generating function
∞
X
(m)
1
∆ − ∆(m−1) · tm
=
(1 − tn1 ) · · · (1 − tnl ) m=0
counts nonnegative integral solutions to the equation a1 n1 + · · · + al nl = m. The
coefficients we are interested in are the sums of the coefficients of this generating
function. Putting n0 := 1, we have
∞
X
1
=
(2)
|∆(m) | · tm .
(1 − tn0 ) · · · (1 − tnl ) m=0
We say n0 , . . . , nl are the affine marks of W .
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LATTICE POINTS IN ALCOVES AND WEYL GROUP EXPONENTS
753
Let n
e := lcm{n0 , . . . , nl }. It is clear that there are polynomials p1 (x), . . . , pne (x)
such that |∆(m) | = pa (m) if m ≡ a (mod n
e) and m > 0. In fact,
1
=
(1 − tn0 ) · · · (1 − tnl )
(3)
l
Y
1 − tne
1 − tni
i=0
n
e
=
!
!
−1
l nX
i
Y
1
(1 − tne )l+1
t
ni k
i=0 k=0
∞ X
µ + l ne µ
t .
l
µ=0
Substituting equation (3) into the left hand side of equation (2), we thus have an
explicit formula for the numbers |∆(m) | and hence for the polynomials pa (x). (It
e} = gcd{b, n
e}. This is not so in general but
turns out that pa (x) = pb (x) if gcd{a, n
holds for the special choices for n0 , . . . , nl we have to consider.)
l
Q
ni , where f is the
Recall that the order of W can be expressed as |W | = f · l! ·
i=1
index of connection, i. e., the determinant of the matrix of Cartan integers. It turns
out that f is the number of affine marks of W equal to 1, i. e., f = |∆(1) | by (2).
l
Q
(mi + 1),
Indeed, this description of f is a consequence of the formula |W | =
i=1
which follows from equation (1), and the formula (4) with m = 1 below.
Observation. Let W be an irreducible finite Weyl group with affine marks
n0 , . . . , nl and exponents m1 , . . . , ml . Define |∆(m) | as in equation (2). Let f be
the index of connection of W and n
e := lcm{n0 , . . . , nl }. Then
|∆(m) | =
(4)
l
f Y
(m + mi )
|W | i=1
for every positive integer m satisfying gcd{m, n
e} = 1.
We are now in the usual situation that an empirically observed result awaits for
a conceptual proof.
3. The tables
The head of each table shows the affine Coxeter-Dynkin diagram. Its vertices
are decorated with the marks n0 , . . . , nl . The left column contains the polynomials
|W |
· |∆(m) |, and the number in the right column is gcd{m, n
e}. The polynomials
f
are ordered by increasing coefficients.
1
s
HH
s s · · · s Hs
bl
A
l
Q
1
1
1
n
e=1
1
(m + i)
i=1
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1
754
RUEDI SUTER
1
s 2
H
Hs
s
bl
B
bl
C
l
Q
i=1
1
1
s> 2s
2
s · · · 2s
2
2
2
s · · · 2s
s> 2s
n
e=2
s< 1s
(m + 2i − 1)
(m + l)
l−1
Q
1
(m + 2i)
2
i=1
1
s 2
H
Hs
s
bl
D
2
s · · · 2s
1
2 s
s
H
Hs
n
e=2
1
1
l−1
Q
(m + l − 1) (m + 2i − 1)
i=1
l−2
Q
2
m + 2(l − 1)m + l(l−1)
(m + 2i)
2
1
2
i=1
1
s 2s 1s
2 s
1 s
(m + 1)(m + 4)(m + 5)(m + 7)(m + 8)(m + 11) =
m6 + 36 m5 + 510 m4 + 3600 m3 + 13089 m2 + 22284 m + 12320
(m + 3)(m + 9)(m4 + 24 m3 + 195 m2 + 612 m + 480) =
m6 + 36 m5 + 510 m4 + 3600 m3 + 13089 m2 + 22284 m + 12960
(m + 2)(m + 4)(m + 8)(m + 10)(m2 + 12 m + 26) =
m6 + 36 m5 + 510 m4 + 3600 m3 + 13224 m2 + 23904 m + 16640
(m + 6)2 (m4 + 24 m3 + 186 m2 + 504 m + 480) =
m6 + 36 m5 + 510 m4 + 3600 m3 + 13224 m2 + 23904 m + 17280
s
b6
E
1
2
s
b7
E
s
2
s
3
s
3
4
s
s
3
s
2
s
n
e=6
1
3
2
6
1
s
n
e = 12
2
(m + 1)(m + 5)(m + 7)(m + 9)(m + 11)(m + 13)(m + 17) =
1
m7 + 63m6 + 1617m5 + 21735m4 + 162939m3 + 663957m2 + 1286963m + 765765
(m + 3)(m + 9)(m + 15)(m4 + 36 m3 + 438 m2 + 2052 m + 2289) =
7
6
5
4
3
3
2
m + 63m + 1617m + 21735m + 162939m + 663957m + 1304883m + 927045
(m + 2)(m + 10)(m + 13)(m + 14)(m3 + 24 m2 + 155 m + 342) =
7
6
5
4
3
2
2
m + 63m + 1617m + 21735m + 163884m + 689472m + 1495808m + 1244880
(m + 4)(m + 5)(m + 8)(m + 16)(m3 + 30 m2 + 263 m + 504) =
4
m7 + 63m6 + 1617m5 + 21735m4 + 163884m3 + 689472m2 + 1495808m + 1290240
(m + 6)(m6 +57 m5 +1275 m4 +14085 m3 +79374 m2 +213228 m+234360) =
6
m7 + 63m6 + 1617m5 + 21735m4 + 163884m3 + 689472m2 + 1513728m + 1406160
(m + 12)(m6 +51 m5 +1005 m4 +9675 m3 +47784 m2 +116064 m+120960) =
7
6
5
4
3
2
m + 63m + 1617m + 21735m + 163884m + 689472m + 1513728m + 1451520
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12
s
2
s
4
3
s
s
6
s
5
s
4
s
3
s
2
s
1
(m + 1)(m + 7)(m + 11)(m + 13)(m + 17)(m + 19)(m + 23)(m + 29) =
m8 + 120 m7 + 6020 m6 + 163800 m5 + 2616558 m4 + 24693480 m3 + 130085780 m2 + 323507400 m + 215656441
(m + 5)(m + 25)(m6 + 90 m5 + 3195 m4 + 56700 m3 + 516183 m2 + 2120490 m + 1948205) =
m8 + 120 m7 + 6020 m6 + 163800 m5 + 2616558 m4 + 24693480 m3 + 130085780 m2 + 323507400 m + 243525625
(m + 3)(m + 9)(m + 21)(m + 27)(m4 + 60 m3 + 1250 m2 + 10500 m + 22749) =
m8 + 120 m7 + 6020 m6 + 163800 m5 + 2616558 m4 + 24693480 m3 + 130802580 m2 + 345011400 m + 348264441
(m + 15)2 (m6 + 90 m5 + 3095 m4 + 50700 m3 + 399183 m2 + 1310490 m + 1671705) =
m8 + 120 m7 + 6020 m6 + 163800 m5 + 2616558 m4 + 24693480 m3 + 130802580 m2 + 345011400 m + 376133625
(m + 2)(m + 14)(m + 22)(m + 26)(m4 + 56 m3 + 1068 m2 + 8344 m + 27176) =
m8 + 120 m7 + 6020 m6 + 163800 m5 + 2626008 m4 + 25260480 m3 + 141860480 m2 + 418876800 m + 435250816
(m + 10)(m7 + 110 m6 + 4920 m5 + 114600 m4 + 1480008 m3 + 10460400 m2 + 37256480 m + 46312000) =
m8 + 120 m7 + 6020 m6 + 163800 m5 + 2626008 m4 + 25260480 m3 + 141860480 m2 + 418876800 m + 463120000
(m + 4)(m + 8)(m + 16)(m + 28)(m4 + 64 m3 + 1428 m2 + 12536 m + 36056) =
m8 + 120 m7 + 6020 m6 + 163800 m5 + 2626008 m4 + 25260480 m3 + 141860480 m2 + 424320000 m + 516898816
(m + 20)(m7 + 100 m6 + 4020 m5 + 83400 m4 + 958008 m3 + 6100320 m2 + 19854080 m + 27238400) =
m8 + 120 m7 + 6020 m6 + 163800 m5 + 2626008 m4 + 25260480 m3 + 141860480 m2 + 424320000 m + 544768000
(m + 6)(m + 18)(m6 + 96 m5 + 3608 m4 + 66840 m3 + 632184 m2 + 2869344 m + 5437152) =
m8 + 120 m7 + 6020 m6 + 163800 m5 + 2626008 m4 + 25260480 m3 + 142577280 m2 + 440380800 m + 587212416
m8 + 120 m7 + 6020 m6 + 163800 m5 + 2626008 m4 + 25260480 m3 + 142577280 m2 + 440380800 m + 615081600
(m + 12)(m + 24)(m6 + 84 m5 + 2708 m4 + 42120 m3 + 329784 m2 + 1257696 m + 2322432) =
m8 + 120 m7 + 6020 m6 + 163800 m5 + 2626008 m4 + 25260480 m3 + 142577280 m2 + 445824000 m + 668860416
m8 + 120 m7 + 6020 m6 + 163800 m5 + 2626008 m4 + 25260480 m3 + 142577280 m2 + 445824000 m + 696729600
b8
E
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60
30
12
6
20
4
10
2
15
3
5
1
n
e = 60
LATTICE POINTS IN ALCOVES AND WEYL GROUP EXPONENTS
755
756
RUEDI SUTER
b4
F
n
e = 12
s 2s 3s> 4s 2s
4
3
2
(m + 1)(m + 5)(m + 7)(m + 11) = m + 24 m + 190 m + 552 m + 385
1
(m + 3)(m + 9)(m2 + 12 m + 19) = m4 + 24 m3 + 190 m2 + 552 m + 513
3
2
4
3
2
(m + 2)(m + 10)(m + 12 m + 44) = m + 24 m + 208 m + 768 m + 880
2
2
2
4
3
2
(m + 6) (m + 12 m + 28) =
m + 24 m + 208 m + 768 m + 1008
6
2
2
4
3
2
(m + 4) (m + 8) =
m + 24 m + 208 m + 768 m + 1024
4
m4 + 24 m3 + 208 m2 + 768 m + 1152
12
1
b2
G
3
s< 2s
2
n
e=6
1
s
(m + 1)(m + 5) = m + 6 m + 5
(m + 2)(m + 4) = m2 + 6 m + 8
(m + 3)2 =
m2 + 6 m + 9
m2 + 6 m + 12
1
2
3
6
4. More observations
The tables suggest more observations. For instance, the polynomials for which
gcd{m, n
e} = 1 have the smallest coefficients, those for which gcd{m, n
e} = n
e the
biggest. This is of course what one expects.
There are |∆(m−h) | points in the “interior” of ∆(m) , i. e., in
λ + ρ λ ∈ P+ , hλ + ρ, θs∨ i < m ,
where ρ =
l
P
i=1
$i . In fact, the points in the “interior” correspond to strictly positive
integral solutions to
a1 n1 + · · · + al nl < m.
(5)
There are |∆
| such solutions because 1 + n1 + · · · + nl = h. (The 1 accounts
for the strict inequality in (5).) |∆(m−h) | is connected to the polynomial for |∆(m) |
by reciprocity, i. e., Serre duality in the context of toric varieties. To be precise, let
e) and m > 0.
pa (x) be the polynomial for which |∆(m) | = pa (m) if m ≡ a (mod n
Then
(m−h)
pa−h (m − h) = (−1)l pa (−m).
In particular, we have 1 = gcd{1, n
e} = gcd{1 − h, n
e}, whence
p1 (m − h) = (−1)l p1 (−m),
or, using the observation (4),
l
Y
i=1
(m − h + mi ) =
l
Y
(m − mi ).
i=1
Hence Serre duality yields a fancy but amusing way of obtaining the symmetry
(h − ml , . . . , h − m1 ) = (m1 , . . . , ml ) of the set of exponents.
The reader may draw further observations from the tables, e. g., describe which
linear factors occur.
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LATTICE POINTS IN ALCOVES AND WEYL GROUP EXPONENTS
757
5. The other affine Coxeter-Dynkin diagrams
bl , F
b4 , and G
b 2 give the number of
bl, C
As already mentioned, the tables for B
dominant integral weights at level m for the twisted affine Kac-Moody algebras.
Physicists may be more interested in the untwisted case and want to count the
number of primary fields for the WZNW conformal field theory models. The tables
for these numbers are given below.
l
b ∨ contains the polynomials l! Q ni |∆(m) |
The left column in the table for X
l
i=0
with |∆(m) | defined as in equation (2) for n0 , . . . , nl as shown in the diagrams, and
the number in the right column is gcd{m, n
e}. As before, n
e = lcm{n0 , . . . , nl }.
Again, if the level m satisfies gcd{m, n
e} = 1, |∆(m) | is a product of l linear factors.
This time the connexion with the exponents is less straightforward.
The sum g := n0 + n1 + · · · + nl is the dual Coxeter number. Now the reciprocity
formula connects |∆(m−g) | with the polynomial for |∆(m) |.
c l which merely counts lattice points for C
b l , but in
Also there is the table for BC
a stuttering manner, meaning that |∆(2µ+1) | = |∆(2µ) | for µ ∈ Z.
6. More tables
1
s 2
H
Hs
s
b∨
B
l
(m +
3
2l
(m + 2l )
− 1)
l−1
Q
l−1
Q
2
s · · · 2s
2
s< 1s
n
e=2
1
(m + 2i − 1)
1
i=1
(m + 2i)
2
i=1
b∨
C
l
1
bl
same table as for A
s< 1s
b∨
F
4
1
s · · · 1s
1
s> 1s
n
e=1
n
e=6
s< 2s 1s
(m + 1)(m + 5)2 (m + 7) =
m4 + 18 m3 + 112 m2 + 270 m + 175
1
3
2
(m + 3)(m + 15 m + 67 m + 69) = m4 + 18 m3 + 112 m2 + 270 m + 207
3
2
4
3
2
(m + 2)(m + 4) (m + 8) =
m + 18 m + 112 m + 288 m + 256
2
3
2
4
3
2
(m + 6)(m + 12 m + 40 m + 48) = m + 18 m + 112 m + 288 m + 288
6
1
s
b∨
G
2
s
1
cl
BC
l
Q
(m + 2i − 1)
3
s> 2s
b2
same table as for B
i=1
l
Q
2
2
s< 2s
2
n
e=2
1
s
s · · · 2s
2
s< 1s
(m + 2i)
i=1
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n
e=2
1
2
758
RUEDI SUTER
References
1. N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5 et 6. Hermann, Paris 1968. Masson,
Paris 1981. MR 39:1590
2. M. Brion, Points entiers dans les polytopes convexes, Séminaire Bourbaki (1993/94). Astérisque
227 (1995), exposé no 780, 145–169. MR 96e:11123
3. K. S. Brown, Buildings. Springer-Verlag, New York–Berlin 1989. MR 90e:20001
4. D. A. Cox, Recent developments in toric geometry. (to appear in the Proceedings of the 1995
Santa Cruz Summer Institute)
5. J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29. Cambridge University Press, Cambridge 1990. MR 92h:20002
Department of Mathematics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139-4307
Current address: Mathematik, Eidgenössische Technische Hochschule Zürich, ETH Zentrum,
8092 Zürich, Switzerland
E-mail address: [email protected]
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