Chapuy and Stump have given a nice generating series for the number of
factorisations of a Coxeter element as a product of reflections. Their method
is to evaluate case by case a character-theoretic expression. The goal of this
note is to give a uniform evaluation of their character-theoretic expression in
the case of Weyl groups, by using combinatorial properties of Deligne-Lusztig
representations.Comment: 5 page

Michel, Jean

English

arXiv

5 pagesChapuy and Stump have given a nice generating series for the number of factorisations of a Coxeter element as a product of reflections. Their method is to evaluate case by case a character-theoretic expression. The goal of this note is to give a uniform evaluation of their character-theoretic expression in the case of Weyl groups, by using combinatorial properties of Deligne-Lusztig representations

Hal-Diderot

”Case-free” derivation for Weyl groups of the number of
reflection factorisations of a Coxeter element
Jean Michel
To cite this version:
Jean Michel. ”Case-free” derivation for Weyl groups of the number of reflection factorisations
of a Coxeter element. 5 pages. 2014. <hal-01052484>
HAL Id: hal-01052484
https://hal.archives-ouvertes.fr/hal-01052484
Submitted on 26 Jul 2014
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destine´e au de´poˆt et a` la diffusion de documents
scientifiques de niveau recherche, publie´s ou non,
e´manant des e´tablissements d’enseignement et de
recherche franc¸ais ou e´trangers, des laboratoires
publics ou prive´s.
“Case-free” derivation for Weyl groups of the
number of reflection factorizations of a Coxeter
element
Jean Michel
8th July 2014
Let W ⊂ GL(Cn) be an irreducible well-generated complex reflection group,
let R be the set of its reflections, R∗ the set of its reflecting hyperplanes, and
let c be a Coxeter element of W (see [CT, Remark 1.3]).
In [CT, Theorem 1.1], Chapuy and Stump obtain a very nice generating
series for the number Nl := |{r1, . . . , rl ∈ Rl | r1 . . . rl = c}| of factorizations of
c into the product of l elements of R. Their formula is
∑
l≥0
tl
l!
Nl =
1
|W |
(et|R|/n − e−t|R
∗|/n)n.
Their method is to obtain a character-theoretic expression for Nl, that they
proceed to evaluate case by case. My observation is that, in the case of Weyl
group, a uniform evaluation of their character-theoretic expression can be done
using properties of Deligne-Lusztig representations. I thank Christian Stump
for making me aware of the problem, and for a careful reading of this text.
We start with [CT, formula above (10)] which states that if S is the element of
the group algebraCW given by
∑
r∈R r, then |W |Nl =
∑
χ∈Irr(W ) χ(1)χ(S
lc−1).
We observe that S is in the center of CW , thus the formula can also be written
|W |Nl =
∑
χ∈Irr(W )
χ(Sl)χ(c−1) (1).
For χ ∈ Irr(W ) write the fake degree of χ as xe1 + . . . + xeχ(1) and define
N(χ∗) =
∑χ(1)
i=1 ei, where χ
∗ denotes the complex conjugate (see for example
[BM, §4.B]). We will need the following property:
Lemma 1. Assume that |CW (H)| has a common value e for all H ∈ R∗. Then
χ(S) = |R|χ(1)−N(χ)−N(χ∗).
Proof. For H ∈ R∗, whose fixator CW (H) is generated by a reflection sH of
hyperplane H and non-trivial eigenvalue ζ = exp(2iπ/e), let mH,i(χ) be the
1
multiplicity of the eigenvalue ζi of sH in the representation affording χ. Then
we get
χ(
e−1∑
j=1
sjH) =
e−1∑
i=0
mH,i(χ)
e−1∑
j=1
ζij = (e− 1)mH,0(χ)−
e−1∑
i=1
mH,i(χ).
Now the formulae in [BM, Corollaire 4.2] read, in our case where |CW (H)| has
a common value e:
∑
H∈R∗
e−1∑
i=1
mH,i(χ) =
N(χ) +N(χ∗)
e
∑
H∈R∗
mH,0(χ) =
|R|χ(1)
e− 1
−
N(χ) +N(χ∗)
e
whence the Lemma.
Remark 2. Another expression valid without assuming |CW (H)| constant is
χ(S) = |R|χ(1) − N(χ) − N(ι(χ)) where ι is Opdam’s involution; to see this,
differentiate with respect to x and then evaluate at x = 1 formula [Ma, 6.5]; see
also [Ma, 6.8].
We now restrict to the case where W is the Weyl group of a connected
reductive algebraic group G over an algebraic closure Fq of the finite field Fq
with q elements. We assume that G is defined over Fq and denote by F the
Frobenius endomorphism defining the corresponding Fq-structure. Let T be an
F -stable maximal torus lying in an F -stable Borel subgroupB. We may identify
W with NG(T) and we assumeG split, which means that F acts trivially onW .
For w ∈ W , let us denote by Rw the (virtual) character of G
F defined by the
Deligne-Lusztig induction RG
Tw
(Id), where Tw is an F -stable maximal torus of
type w (with respect to T). Here Rw is a Qℓ-character, for some prime number
ℓ not dividing q, but we will consider it as a complex character by choosing a
suitable embedding Qℓ →֒ C. The set E(G
F , 1) of constituents of the various
Rw is called the set of unipotent characters of G
F .
The character R1 identifies to that of Ind
G
F
BF
Id, and, since the commuting
algebra of this representation is the Hecke algebra H, isomorphic to the group
algebra of W , we have a decomposition of the form R1 =
∑
χ∈Irr(W ) χ(1)Uχ
where the Uχ are certain characters of G
F called the principal series unipotent
characters.
The set E(GF , 1) and the values 〈Rw, ρ 〉GF for ρ ∈ E(G
F , 1) are inde-
pendent of q; they provide an additional combinatorial structure on W which
can actually be entirely determined by the Hecke algebra. In the case where
W is the symmetric group, the characters Uχ exhaust the set E(GF , 1) and
〈Rw, Uχ 〉GF = χ(w), so Deligne-Lusztig combinatorics bring nothing new. We
could characterize their role in the current proof as enabling the use for other
Weyl groups of the same features which occur in the proof in the symmetric
group case.
2
By linearity we attach to any a ∈ CW a class function Ra on GF , given
if a =
∑
w∈W aww by Ra =
∑
w awRw. In the particular case where a is the
idempotent 1|W |
∑
w∈W χ(w
−1)w attached to χ ∈ Irr(W ) we denote by Rχ the
corresponding class function. Inverting we get Rw =
∑
χ χ(w)Rχ. By [Lu2,
3.19.2] we have 〈Rχ, Rψ 〉GF = δχ,ψ. It follows that for any two elements
a, b ∈ CW we have 〈Ra, Rb 〉GF =
∑
χ∈Irr(W ) χ(a)χ(b). Thus formula (1)
becomes
|W |Nl = 〈RSl , Rc 〉. (1
′)
Lemma 3. We have χ(S)χ(1) = |R|−aχ−Aχ, where aχ (resp. Aχ) is the valuation
(resp. the degree) of the generic degree of H attached to χ (see for example [BM,
§2.B]).
Proof. This is just a translation of Lemma 1 in terms of the invariants coming
from the Hecke algebra. In the case of Coxeter groups in Lemma 1 we have e = 2
and χ = χ∗. Thus Lemma 1 for Coxeter groups becomes χ(S) = χ(1)|R| −
2N(χ). We then conclude by [BM, formula 4.21] which states that in Coxeter
groups N(χ) = χ(1)
aχ+Aχ
2 .
Lemma 4. For ρ ∈ E(GF , 1) we have
〈RSl , ρ 〉GF =
{
χ(Sl) if ρ = Uχ for some χ ∈ Irr(W )
0 otherwise
.
Proof. The blocks of the matrix {〈Rχ, ρ 〉GF }χ∈Irr(W ),ρ∈E(GF ,1) are called the
Lusztig families. They constitute thus a partition Ξ of E(GF , 1) such that for a
“family” F ∈ Ξ we have:
• If Uχ ∈ F , ρ /∈ F then 〈Rχ, ρ 〉GF = 0.
Given a family F , the invariants aχ and Aχ take a constant value on the χ ∈
Irr(W ) such that Uχ ∈ F (see, for example [Lu3, 4.23, 5.25 and 5.27]), thus
χ(S)/χ(1) takes a constant value that we will denote cF on a family F . Thus,
for ρ ∈ F we have:
〈RSl , ρ 〉GF = 〈
∑
χ∈Irr(W )
χ(Sl)Rχ, ρ 〉GF =
∑
{χ|Uχ∈F}
χ(Sl)〈Rχ, ρ 〉GF
= clF
∑
{χ|Uχ∈F}
χ(1)〈Rχ, ρ 〉GF = c
l
F 〈
∑
χ∈Irr(W )
χ(1)Rχ, ρ 〉GF
=
{
0 unless ρ is a Uχ
clFχ(1) = χ(S
l) if ρ = Uχ
Lemma 5. If c is a Coxeter element and χ ∈ Irr(W ) then 〈Rc, Uχ 〉GF = 0
unless χ is an exterior power of the reflection character ref of W . Moreover
〈Rc, U∧i ref 〉GF = (−1)
i.
3
Proof. Let Xc be the Deligne-Lusztig variety defining Rc, so that for g ∈ GF
we have Rc(g) =
∑
i(−1)
iTrace(g | Hic(Xc,Qℓ)). By [Lu1, 6.7 (ii)] applied with
I = ∅ and R0 = Id, we know that in each cohomology group Hic(Xc,Qℓ) with
i = n, n+ 1, . . . , 2n, there is exactly one irreducible representation UχW
i
in the
principal series and that it has multiplicity 1. The fact that χWi = ∧
2n−i ref
is [Lu1, remark 7.8] applied with I = ∅ and R0 = Id. In this case the Hecke
algebra H of loc. cit. is the same as our algebra H and the i-th power of the
reflection representation of H defines precisely the character U∧i ref.
Using Lemmas 4 and 5 to evaluate (1′) we get:
|W |Nl =
n∑
i=0
(−1)i ∧i ref(Sl). (1′′)
Lemma 6. For any well-generated irreducible complex reflection group, formula
(1′′) is equivalent to the Chapuy-Stump formula.
Proof. Since the representations ∧i ref are irreducible by a result of Steinberg
(see [Bou, §2 ex. 3]), formula (1′′) can be written
|W |Nl =
n∑
i=0
(−1)i(∧i ref(S)/ ∧i ref(1))l ∧i ref(1).
Let us pick a reflecting hyperplane H , let e = |CW (H)| and ζ = exp(2iπ/e)
and let s be a reflection with eigenvalue ζ which generates CW (H). Let us
compute
∑j=e−1
j=1 ∧
i ref(sj). We may choose a basis e1, . . . , en of C
n such that
ei ∈ H for i = 1, . . . , n − 1 and sen = ζen. A basis of ∧iCn consists of
eI = ea1∧. . .∧eai where I = {a1, . . . , ai} with a1 < . . . < ai runs over all subsets
of {1, . . . , n} of cardinality i. We have
∑e−1
j=1 s
j(eI) =
{
(e− 1)eI if n /∈ I
−eI otherwise
whence
∑j=e−1
j=1 ∧
i ref(sj) = (e − 1)
(
n
i
)
− e
(
n−1
i−1
)
and ∧i ref(S) = |R|(
(
n
i
)
−(
n−1
i−1
)
)−|R∗|
(
n−1
i−1
)
. We finally get ∧i ref(S)/∧i ref(1) = |R|(1− in )−|R
∗| in and
|W |Nl =
n∑
i=0
(−1)i(|R|(1 −
i
n
)− |R∗|
i
n
)l
(
n
i
)
,
which is exactly what one gets when expanding the Chapuy-Stump formula.
Note. I believe that Lemmas 4 and 5 hold for any Spetsial complex reflection
group, with an appropriate definition of a formal set of unipotent characters
(see [BMM]). I checked it by computer for the primitive irreducible Spetsial
complex reflection groups using [Mi].
For Spetsial groups whose reflections have order 2 the proof of Lemma 4
remains formally valid since Lemma 3 remains true for such groups, thus also
the fact that χ(S)/χ(1) is constant on families. When reflections do not all have
order 2, there are examples where χ(S)/χ(1) is not constant on families so the
proof of Lemma 4 has to change.
4
References
[Bou] N. Bourbaki, “Groupes et alge`bres de Lie”, Chap 5, Hermann 1969.
[BM] M. Broue´ et J. Michel, “Sur certains e´le´ments re´guliers des groupes de
Weyl et les varie´te´s de Deligne–Lusztig associe´es”, Progress in Mathe-
matics no 141 (Birkhauser 1996) 73–139.
[BMM] M. Broue´, G. Malle and J. Michel, “Split Spetses for primitive reflection
groups”,Aste´risque 359 (2014) 1–146.
[CT] G. Chapuy and C. Stump,“Counting factorizations of Coxeter elements
into products of reflections”, arXiv:1211.2789, to appear in Journal of
London Math. Soc.
[Lu1] G. Lusztig, “Coxeter Orbits and Eigenspaces of Frobenius”, Inventiones
38(1976), 101–159.
[Lu2] G. Lusztig, “Representations of finite Chevalley groups”, Regional Conf.
Series in Math. 39 (1978) AMS, 48p.
[Lu3] G. Lusztig, “Characters of reductive groups over a finite field”, Ann.
Math. Studies 107, Princeton UP (1984) 384p
[Ma] G. Malle,“On the rationality and fake degrees of characters of cyclotomic
algebras”, J. Math. Sci. Univ. Tokyo 6(1999) 647–677.
[Mi] J. Michel,“The development version of the CHEVIE package of GAP3.”,
arXiv:1310.7905 [math.RT].
5


"Case-free" derivation for Weyl groups of the number of reflection factorisations of a Coxeter element

https://hal.archives-ouvertes.fr/hal-01052484/document

"Case-free" derivation for Weyl groups of the number of reflection factorisations of a Coxeter element

Abstract

Similar works

Full text

Available Versions

Hal-Diderot