An inductive approach to Coxeter arrangements and Solomon’s descent algebra

DOI 10.1007/s10801-011-0301-9

An inductive approach to Coxeter arrangements and Solomon’s descent algebra

J. Matthew Douglass·Götz Pfeiffer· Gerhard Röhrle

Received: 4 April 2011 / Accepted: 13 June 2011 / Published online: 20 July 2011

© Springer Science+Business Media, LLC 2011

Abstract In our recent paper (Douglass et al.arXiv:1101.2075(2011)), we claimed that both the group algebra of a finite Coxeter groupWas well as the Orlik–Solomon algebra ofW can be decomposed into a sum of induced one-dimensional represen- tations of centralizers, one for each conjugacy class of elements ofW, and gave a uniform proof of this claim for symmetric groups. In this note, we outline an in- ductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for finite Coxeter groups of rank up to 2.

Keywords Coxeter groups·Reflection arrangements·Descent algebra·Dihedral groups

1 Introduction

LetWbe a finite Coxeter group, generated by a setSof simple reflections. If|S| =r, thenWacts as a reflection group on Euclideanr-spaceV. The reflection arrangement ofW is the hyperplane arrangement consisting of the reflecting hyperplanes inV of all the reflections inW. The Orlik–Solomon algebraA(W )ofW is the cohomology

J.M. Douglass

Department of Mathematics, University of North Texas, Denton, TX 76203, USA e-mail:douglass@unt.edu

G. Pfeiffer

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland

e-mail:goetz.pfeiffer@nuigalway.ie G. Röhrle (

Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany e-mail:gerhard.roehrle@rub.de

ring of the complement of the complexified reflection arrangement. It follows from a result of Brieskorn [3] that the algebraA(W )is aW-module of dimension|W|. For some history of the computation ofA(W )as aW-module, see the introduction of our recent paper [4].

In [4], we claimed that both the group algebra CW of W (affording the regular characterρ_W) as well as the Orlik–Solomon algebra A(W ) (affording the Orlik–

Solomon characterω_W) can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements ofW, in the following interlaced way.

Conjecture A LetRbe a set of representatives of the conjugacy classes ofW. Then, for eachw∈R, there are linear charactersϕ_wandψ_w ofC_W(w)such that

ρW=

w∈R

Ind^W_C

W(w)ϕw, ωW=

w∈R

Ind^W_C

W(w)ψw

are sums of induced linear characters. Moreover, for eachw∈R, the charactersϕ_w andψ_wcan be chosen so that

ψ_w=ϕ_wα_w,

whereis the sign character ofW, andα_w is the determinant on the 1-eigenspace ofw.

WhenWis a symmetric group, the formula forρ_W has been proved independently by Bergeron, Bergeron, and Garsia [1], Hanlon [6], and Schocker [14]. The formula forω_Wfollows from work of Lehrer and Solomon [9], who also checked the identity forω_Win the case of a dihedral groupW. Conjecture 2.1 in [4] is a graded refinement of ConjectureAand the main result in [4] is a uniform proof of this refined conjecture for symmetric groups.

The details of the proof of Conjecture 2.1 in [4] for symmetric groups rely on properties of these groups not shared by other finite Coxeter groups. However, the underlying strategy of the proof using induced characters both generalizes and admits a “relative” version, for pairs (W, W_L), where W_L is a parabolic subgroup of W. In Sect.4, we formalize this notion in ConjectureC, show how it leads to a proof of Conjecture A, and describe a two-step procedure that can be used to prove this relative conjecture. Prior to that, in Sects.2and3we review some notation and basic facts about the descent algebraΣ (W )and the Orlik–Solomon algebraA(W ). In the final section, we apply the methods from Sect.4and prove ConjectureCfor all pairs (W, WL)whereW is arbitrary and WLhas rank at most 2. As a consequence, we deduce that Conjecture Aholds for Coxeter groups of rank 2 or less.

2 Minimal length transversals of parabolic subgroups

The descent algebra of a finite Coxeter groupW encodes many aspects of the com- binatorics of the minimal length coset representatives of the standard parabolic sub- groups ofW. In this section, we provide notation and summarize useful properties of these distinguished coset representatives following Pfeiffer [12].

ForJ⊆S, let

XJ=

w∈W: (sw) > (w)for alls∈J .

ThenX_J is a right transversal of the parabolic subgroupW_J= JofW, consisting of the unique elements of minimal length in their cosets. If we set

xJ=

x∈XJ

x⁻¹∈CW,

then, by Solomon’s Theorem [15], the subspace Σ (W )= x_J :J⊆S_C

is a 2^r-dimensional subalgebra of the group algebraCW, called the descent algebra ofW.

ForJ⊆S, denote

X_J=

x∈X_J :J^x⊆S .

The action ofWon itself by conjugation partitions the power set ofSinto equivalence classes ofW-conjugate subsets. We call the class

[J] =

J^x:x∈X_J of a subsetJ⊆Sthe shape ofJ, and denote by

Λ=

[J] :J⊆S

the set of shapes ofW. The shapes parametrize the conjugacy classes of parabolic subgroups ofW, since two subsetsJ, K⊆S are conjugate if and only if the corre- sponding parabolic subgroupsW_J andW_K are conjugate. We say that a parabolic subgroup ofW has shape[J]if it is conjugate toW_J inW.

Furthermore, forJ⊆S, we define N_J =

x∈X_J:J^x=J .

ThenNJ is a subgroup ofW and by results of Howlett [7], the normalizer ofWJ in Wis a semi-direct productN_W(W_J)=W_JN_J.

An elementw∈Wis called cuspidal in casewhas no fixed points in the reflection representation ofW. Thus, forJ⊆S, an elementw∈W_Jis cuspidal in the parabolic subgroupW_J ifwhas no fixed points in the orthogonal complement of Fix(WJ)inV, where Fix(WJ)is the fixed point subspace ofW_J inV. Ifw is a cuspidal element inWJ, then the quotientCW(w)/CW_J(w)is isomorphic toNJ (see [8]).

We consider the characterαJ ofNW(WJ), defined, forw∈NW(WJ), as α_J(w)=det(w|Fix(W_J)).

Note thatW_J is contained in the kernel ofα_J and soα_J(un)=α_J(n) foru∈W_J, n∈N_J.

Lemma 2.1 LetJ ⊆S. For n∈N_J denote by σ_J(n) the sign of the permutation induced onJby conjugation withn. Then

σJ(n)=(n)αJ(n), for alln∈NJ.

Proof Denote byVJ the orthogonal complement of Fix(WJ)inV. ThenVJ affords the reflection representation of the parabolic subgroupW_J, and the decomposition V =VJ⊕Fix(WJ)isNW(WJ)-stable. Forn∈NJ, the matrix ofnonVJ is equiv- alent to the permutation matrix of the conjugation action ofnonJ and thus has de- terminantσJ(n). The matrix ofnon Fix(WJ)has determinantαJ(n), by definition.

Consequently, the determinant ofnonV is(n)=σ_J(n)α_J(n).

Pfeiffer and Röhrle [13] callWJ a bulky parabolic subgroup ofW ifNW(WJ)is isomorphic to the direct productW_J ×N_J, or equivalently, if N_J centralizesW_J. Notice thatWJ is bulky wheneverWJ is a self-normalizing subgroup ofW. Suppose W_J is bulky inW. Then σ_J(n)=1 for all n∈N_J. Consequently, for u∈W_J and n∈NJ, we have

(un)α_J(un)=(u). (2.2)

Thus, the characterαJ =J×1N_J ofNW(WJ)=WJ×NJ is the trivial extension of the sign character ofW_J.

Here and in the remainder of the paper, we denote the restrictions of the trivial and the sign character ofW to a subgroupU ofW by 1U and_U, respectively, or by 1J and_J, ifU=W_J for someJ⊆S. If no confusion can arise, we denote the restrictions of the characters 1S and_S ofW to any of its subgroups simply by 1 and, respectively.

Following Bergeron et al. [2], we decomposeΣ (W )into projective indecompos- able modules, using a basis of quasi-idempotents, that naturally arise as follows. For L, K⊆S, we define

m_KL=

|XK∩X_L|, ifL⊆K,

0, otherwise.

Then(mKL)K,L⊆Sis an invertible matrix, and consequently, there is a basis(eL)L⊆S

ofΣ (W )such that

xK=

L⊆S

mKLeL

forK⊆S. Define, forλ∈Λ, elements e_λ=

L∈λ

e_L.

Then{e_λ:λ∈Λ}is a set of primitive, pairwise orthogonal idempotents inΣ (W ). In particular,

λ∈Λ

e_λ=1∈CW.

Thus, if we set

Eλ=eλCW, then

CW=

λ∈Λ

Eλ (2.3)

is a decomposition of the group algebra into right ideals. We call the right idealE_[S]

the top component ofCW.

Forλ∈Λ, denote by Φ_λ the character of the W-moduleE_λ. Furthermore, for L⊆S, denote byΦ_Lthe character of the top component of the group algebraCW_L. Notice that forλ= [L],Φ_[L]is a character ofW whereasΦ_Lis a character ofW_L. IfL=S, thenW_L=W andΦ_[S]=Φ_S. In general, the charactersΦ_[L]andΦ_Lare related in the following way.

Proposition 2.4 [4, Corollary 3.13] LetL⊆S. Then the characterΦ_LofW_Lextends to a characterΦ_Lof the normalizerN_W(W_L)=W_LN_Lsuch that

Φ_[L]=Ind^W_N

W(WL)ΦL.

Remark 2.5 IfW_Lis a bulky parabolic subgroup ofW, then, by [4, Lemma 3.7]Φ_L is the characterΦ_L×1NLofN_W(W_L)=W_L×N_Land soΦ_[L]=Ind^W_W

L×N_L(Φ_L× 1NL).

3 The reflection arrangement and the Orlik–Solomon algebraA(W )

A finite Coxeter group of rankr acts as a reflection group on Euclidean spaceR^r. Here it is convenient to regard this as an action on the complex spaceV_C=C^r. Let

T =

s^w:s∈S, w∈W

be the set of reflections ofW. Fort∈T, denote byHt the reflecting hyperplane of t, i.e., the 1-eigenspace of t. The set of hyperplanesA= {Ht :t ∈T}is called the reflection arrangement ofW; for details see [11, Chap. 6]. Examples of (the real part of) reflection arrangements in dimension 2 are shown in Figs.1and2.

The lattice ofAis the set of all possible intersections of hyperplanes L(A)= {H_t1∩ · · · ∩H_tp:t1, . . . , t_p∈T}.

ForX∈L(A), the pointwise stabilizer

WX= {w∈W:x.w=xfor allx∈X}

is a parabolic subgroup ofW. We define the shape sh(X)ofXto be the shape ofW_X, i.e., sh(X)= [L] ∈ΛifW_Xis conjugate toW_LinWfor someL⊆S. The groupW acts onT by conjugation and theW-action onT induces actions of W onAand

Fig. 1 Hyperplane arrangements of typeI₂(m),m=3,5,7,9

Fig. 2 Hyperplane arrangements of typeI2(m),m=4,6,8,10

L(A). Orlik and Solomon [10] have shown that the normalizer ofW_X inW is the setwise stabilizer ofXinW, that is,

N_W(W_X)= {w∈W:X.w=X}.

Consequently, the orbits ofW on the latticeL(A)are parametrized by the shapes of W. We denote byαX:NW(WX)→Cthe linear character ofNW(WX)defined by

αX(w)=det(w|X)

for w∈N_W(W_X). Then, for w∈W, we have α_w =α_X, where X=Fix(w), the fixed point subspace of w in V. Moreover, for L⊆S, we have α_L=α_X, where X=Fix(WL).

The Orlik–Solomon algebra ofW is the associativeC-algebraA(W ), generated as an algebra by elementsa_t,t∈T, subject to the relations

a_ta_t = −a_ta_t for allt, t ∈T, and

p i=1

(−1)ⁱat₁· · ·at_i−1at_iat_i+1· · ·at_p=0,

where the hat denotes omission, whenever{H_t1, . . . , H_tp}is linearly dependent. The action ofW on the hyperplanes extends to an action onA(W )via

a_t.w=a_t^w

fort∈T,w∈W. The algebraA(W )is a skew-commutative, graded algebra A(W )=

p≥0

A^p,

where the degreepsubspaceA^pis spanned by those monomialsa_t1· · ·a_tpinA(W ) with dimH_t1 ∩. . .∩H_tp=r−p. Clearly,A^p=0 forp > r. We call A^r the top component ofA(W ). We need a refinement of this decomposition, due to Brieskorn [3]. For a subspaceX∈L(A)of codimensionp, define a subspace

A_X= a_t1· · ·a_tp:H_t1∩. . .∩H_tp=X

ofA(W ). ThenA_{0}=A^ris the top component ofA(W ). Note thatAXis an embed- ding of the top component ofA(W_X)intoA(W ). Forw∈W, we haveA_X.w=A_X.w and soA_Xis anN_W(W_X)-stable subspace.

We have

A(W )=

X∈L(A)

A_X

and if we set

A_λ=

sh(X)=λ

A_X,

forλ∈Λ, then

A(W )=

λ∈Λ

Aλ

is a decomposition ofA(W )intoW-modulesA_λ. Note thatA_[S]=A_{0} is the top component ofA(W ).

Forλ∈Λ, denote byΨλthe character of the componentAλof the Orlik–Solomon algebraA(W ). Furthermore, forL⊆S, denote byΨLthe character of the top com- ponent of the Orlik–Solomon algebraA(WL)of the parabolic subgroupWL ofW. Notice that forλ= [L],Ψ_[L] is a character ofW whereasΨLis a character ofWL. IfL=S, thenΨ_[S]=ΨS. In general, the charactersΨ_[L] andΨL are related in the following way, analogous to Proposition2.4.

Proposition 3.1 [9, §2] LetL⊆S. Then the characterΨLofWLextends to a char- acterΨLof the normalizerNW(WL)=WLNLsuch that

Ψ_[L]=Ind^W_N

W(WL)ΨL.

Remark 3.2 Suppose that W_L is a bulky parabolic subgroup of W and set X= Fix(WL). If codimX=p and t₁, . . . , t_p are in T withX=H_t1 ∩ · · · ∩H_tp, then t1, . . . , tp are in WL and so, since NL centralizes WL, we have at₁· · ·at_p.n= a_tⁿ

1 · · ·at_pⁿ=at₁· · ·at_p, forn∈NL. Thus,ΨLis the characterΨL×1N_LofNW(WL)= WL×NLand soΨ_[L]=Ind^W_W

L×NL(ΨL×1N_L).

4 The inductive strategy

Before stating our relative Conjecture C, we briefly review the proof of Conjec- ture 2.1 in [4] and describe how it leads to a proof of ConjectureA. We first showed that the characters of the top components ofCWandA(W )are related as described in the following conjecture which makes sense for any finite Coxeter group. To this end, letCbe the set of cuspidal conjugacy classes ofWand, forL⊆S, letCLdenote the set of cuspidal conjugacy classes inW_L. For a classCinCorCL, we denote by w_C∈Ca fixed representative.

Conjecture B For each classC∈C, there exist linear charactersϕ_wC andψ_wC of the centralizerC_W(w_C)such that the following hold:

(i) ΦS= C∈CInd^W_C

W(w_C)ϕw_C; (ii) Ψ_S= _C∈CInd^W_C

W(wC)ψ_wC; (iii) ψ_wC=ϕ_wCfor allC∈C.

Remark 4.1 If it is known thatΨS=ΦSS, then choosingψw_C orϕw_C in such a way thatψw_C=ϕw_C, we have that part (iii) in the above ConjectureBholds and that (i) and (ii) are equivalent statements.

WhenWis a symmetric group, every parabolic subgroupW_LofWis a product of symmetric groups and so ConjectureBholds for the groupW_L. Thus, forw_C∈C∈ CL, we obtained linear charactersϕ_wC andψ_wC ofC_WL(w_C)such that the characters Φ_LandΨ_LofW_Ldecompose as

ΦL=

C∈CL

Ind^W_C^L

WL(w_C)ϕw_C and ΨL=

C∈CL

Ind^W_C^L

WL(w_C)ψw_C.

We know from Propositions2.4and3.1thatΦ_LandΨ_Lextend to charactersΦ_Land Ψ_LofN_W(W_L). The next step in [4] was to show that eachϕ_wC andψ_wC extend to charactersϕ_wC andψ_wC ofC_W(w_C)in such a way that

ΦL=

C∈CL

Ind^N_C^W(WL)

W(w_C)ϕw_C and ΨL=

C∈CL

Ind^N_C^W(WL)

W(w_C)ψw_C, (4.2) and moreover thatψw_C =ϕw_CSαL for all C∈CL. Finally, we applied Ind^W_N

W(W_L)

to (4.2) and summed over the set of shapes [L] ∈Λ. Conjecture Athen followed immediately by transitivity of induction.

Motivated by (4.2), we make the following general conjecture.

Conjecture C LetL⊆S. Then, for eachC∈CL, there exist linear charactersϕ_wC andψ_wC ofC_W(w_C)such that the following hold:

(i) Φ_L= _C∈CLInd^N_C^W(WL)

W(wC)ϕ_wC; (ii) Ψ_L= _C∈CLInd^N_C^W(WL)

W(w_C)ψ_wC; (iii) ψ_wC=ϕ_wCSα_Lfor allC∈CL.

Remark 4.3 If it is known thatΨ_L=Φ_LSα_L, then choosing ψ_wC orϕ_wC in such a way thatψ_wC=ϕ_wCSα_L, we have that part (iii) in the above ConjectureCholds and that (i) and (ii) are equivalent statements.

ConjectureBis known to hold in the following cases:

1. Wof typeA(see [4, Theorem 4.1]);

2. Whas rank 2 or less (see Lemmas5.1and5.2, Theorem5.11).

ConjectureCis known to hold in the following cases:

1. Wof typeA; allL(see [4, Theorem 5.2]);

2. Warbitrary;WLis bulky and satisfies ConjectureB(by Theorem4.7);

3. Warbitrary;|L| ≤2 (see Corollary5.3, Theorem5.18).

If ConjectureCholds for allL⊆S, then ConjectureAis true forW.

Theorem 4.4 Suppose that Conjecture Cholds for all subsetsL⊆S. Then for each win a setRof representatives of the conjugacy classes ofW, there are linear char- actersϕ_w andψ_wofC_W(w)such that

(i) the regular character ofW is given byρW= w∈RInd^W_C

W(w)ϕw, (ii) the Orlik–Solomon character ofWis given byωW= _w∈RInd^W_C

W(w)ψw, and (iii) ψw=ϕwαw for allw∈R.

Proof ForL⊆S, letRL be a set of representatives of the classesCL. For a class C∈CL, denote byw_C∈RL its representative. LetLbe a set of representatives of shapes, soΛ= { [L] |L∈L}. Then, by [5, Theorem 3.2.12], we may assume without loss that

R=

L∈L

RL= {w_C:C∈CL, L∈L}.

Then, by ConjectureCthe equality in (iii) holds. By (2.3) and Proposition2.4, we have

ρ_W=

λ∈Λ

Φ_λ=

L∈L

Ind^W_N

W(W_L)Φ_L=

L∈L

C∈CL

Ind^W_C

W(w_C)ϕ_wC,

as desired. The formula forωWfollows in the same way.

Notice that in the case whenL=S, ConjectureCis simply a restatement of Con- jectureB. In general, ConjectureCforL⊆Simplies the validity of ConjectureBfor the groupW_L, as follows.

Proposition 4.5 Suppose that ConjectureCholds for a subsetL⊆S. Then the re- strictions

ϕ_wC=Res^C_C^W(wC)

WL(wC)ϕ_wC and ψ_wC=Res^C_C^W(wC)

WL(wC)ψ_wC are linear characters that satisfy ConjectureBforW_L.

Proof By Mackey’s theorem, we have Res^N_W^W(WL)

L Ind^N_C^W(WL)

W(wC)ϕ_wC =Ind^W_C^L

WL(wC)Res^C_C^W(wC)

WL(wC)ϕ_wC, sinceN_W(W_L)=W_LC_W(w_C)(see [8]), and therefore,

Φ_L=Res^N_W^W(WL)

L Φ_L

=

C∈CL

Res^N_W^W(WL)

L Ind^N_C^W(WL)

W(w_C)ϕ_wC

=

C∈CL

Ind^W_C^L

WL(w_C)Res^C_C^W(wC)

WL(w_C)ϕw_C

=

C∈CL

Ind^W_C^L

WL(wC)ϕ_wC.

The formula forΨ_Lfollows in the same way. The conclusion thatψ_wC =ϕ_wCfor

C∈CLis easily seen to hold.

Remark 4.6 Although Conjecture BforW_Lformally follows from ConjectureC, as in [4], the charactersϕ_wC andψ_wC ofC_W(w_C)arise in practice as extensions of charactersϕ_wC andψ_wC ofC_WL(w_c)that satisfy ConjectureBforW_L. In particular, if ConjectureBis known to hold forW_L, then using Remark4.3, to prove Conjec- tureCforL⊆S, it suffices to prove that eachϕ_wCextends toC_W(w_C)in such a way that ConjectureC(i) holds and thatΨ_L=Φ_LSα_L.

WhenL⊆Sis such thatW_Lis a self-normalizing subgroup ofW(e.g., ifL=S), thenN_Lis the trivial group and ConjectureBfor the groupW_L vacuously implies ConjectureCfor the subset Lin this case. More generally, whenever the comple- mentNLcentralizesWL, i.e., whenWLis bulky inW, ConjectureBforWLimplies ConjectureCforL⊆S, as follows.

Theorem 4.7 Let L⊆S. Suppose that ConjectureB holds for the groupWL and thatWL is a bulky parabolic subgroup ofW. Then ConjectureCholds withϕw_C = ϕw_C ×1N_L andψw_C=ψw_C×1N_Lfor each cuspidal classCofWL.

Proof As observed in the remark above, it suffices to show that eachϕ_wC extends to C_W(w_C)in such a way that ConjectureC(i) holds and thatΨ_L=Φ_LSα_L.

BecauseN_L centralizesW_L, we have that the centralizerC_W(w_C)is the direct product ofC_WL(w_C), and N_L and soϕ_wC is indeed a linear character ofC_W(w_C) that extendsϕ_wC. Thanks to Remark2.5,Φ_L=Φ_L×1NL. Thus, by ConjectureB(i), we have

Φ_L=Φ_L×1NL=

C∈CL

Ind^W_C^L

WL(w_C)ϕ_wC

×1NL

=

C∈CL

Ind^W_C^L×NL

WL(w_C)×N_L(ϕw_C ×1N_L)

=

C∈CL

Ind^N_C^W(WL)

W(wC)ϕ_wC. Hence ConjectureC(i) holds.

By Remark3.2, ConjectureB(iii), Lemma2.1, and Remark2.5, we have ΨL=ΨL×1N_L=ΦLL×1N_LσL=(ΦL×1N_L)αL=ΦLSαL,

using the fact thatWL⊆kerαL, whence we are done.

Combining Theorem4.7with the results in [4], we see that ifWLis a product of Coxeter groups of typeAand is a bulky parabolic subgroup ofW, then Conjecture Cholds forL⊆S. For example, if WL is of typeA1×A3 andW is of typeE6, then the charactersϕw_C andψw_C constructed in [4] satisfy Conjecture Band so, by Theorem4.7, they extend toC_W(w_C)and Conjecture Cholds. Note, however, that the property of being a bulky parabolic subgroup depends in a fundamental way on the embedding ofW_LinW. IfW_Lis of typeA₁×A₃andW is of typeE₇, thenW_L is not bulky and Theorem4.7cannot be applied.

5 ConjecturesA,BandCfor Coxeter groups of rank up to 2

In this section, we show that Conjecture Cholds for L⊆S for anyS as long as

|L| ≤2. Note that because the type of the ambient Coxeter groupWis arbitrary, even for typesA1×A1 andA2Conjecture Cis a stronger statement than is proved in [4] for such parabolic subgroups. The strategy we use is to first prove that Conjec- ture B holds forW when the rank of W is at most 2 and then use the procedure outlined in Remark4.6. Combining Conjecture Cwith Theorem4.7, we conclude that Conjectures A, B, and Call hold in case the rank ofW is at most 2.

The top components of Coxeter groups of rank 0 or 1 almost trivially satisfy Con- jectureB. For later reference, we record this explicitly in the following lemmas.

Lemma 5.1 The top component characters ofW_∅ are Φ_∅=1_∅ and Ψ_∅=1_∅. Moreover,W_∅satisfies ConjectureBwithϕ₁=1_∅andψ₁=1_∅.

Lemma 5.2 SupposeW is a Coxeter group of rank 1, generated byS= {s}. Then the top component characters ofW areΦ_S=_SandΨ_S=1S. Moreover,Wsatisfies ConjectureBwithϕ_s=_S andψ_s=1S.

Proof In this case, the non-trivial conjugacy class{s}is the unique cuspidal conju- gacy class inW. From the definitions we havee_[S]=e_S=¹₂(1−s)and it follows that

Wacts on the top componentE_[S]=e_[S]CWwith characterΦ_[S]=_S. Moreover,W acts trivially on the basis{a_s}of the top componentA_[S]ofA(W ), which therefore affords the trivial character. Thus,Ψ_[S]=1S and soΦ_[S]=Ψ_[S]S. Setϕs =S and ψs=1S. Thenϕsandψs obviously satisfy the conclusions of ConjectureB.

In any finite Coxeter groupW, parabolic subgroups of rank 0 and 1 are always bulky. We may thus conclude from Lemmas5.1and5.2and Theorem4.7that Con- jecture Cholds forL⊆Swith|L| ≤1.

Corollary 5.3 Suppose thatL⊆Shas size|L| ≤1. Then ConjectureCholds.

As a consequence of the corollary,W acts trivially on both the componentE_[∅]

of the group algebraCW(with characterΦ_[∅]=Φ_∅=1S) and the componentA_[∅]

of the Orlik–Solomon algebraA(W )(with characterΨ_[∅]=Ψ_∅=1S), as one can easily establish directly.

Moreover, the degree 1 component ofA(W )is a direct sum of transitive permuta- tion modules, one for each conjugacy class of reflections ofW. This agrees with the description of the degree 1 component ofA(W )as the permutation representation of Won its reflections, that can easily be obtained directly.

Next we consider the case whenW has rank 2. Until further notice, we assume that

W=

s, t:s²=t²=(st )^m=1 .

ThenWis a Coxeter group of rank two and is of typeA₁×A₁, orI₂(m)form≥3, with Coxeter generatorsS= {s, t}. For convenience, we regard typeA1×A1as type I2(2), noting that the general results of this section remain true form=2.

To prove ConjectureBforW, we first compute the characterΦS of the top com- ponentE_[S]of the group algebraCW, and verify that it is a sum of induced linear characters. Then we compute the characterΨ_S of the top component A_[S] of the Orlik–Solomon algebraA(W )and verify thatΨ_S =Φ_SS. Conjecture Bthen fol- lows as observed in Remark4.1.

As usual, denote byw₀the longest element ofW. Furthermore, we define Av(U )= 1

|U|

u∈U

u

for a subgroup U of W. Recall that Av(U )u=Av(U ) for all u ∈U and that Av(U )CW is the permutation module ofW on the cosets ofU.

Lemma 5.4 eS=Av(w0)−Av(W ).

Proof By Solomon’s theorem [15], the elements

x_∅=1+s+t+st+t s+ · · · +w₀, x_s=1+t+st+t st+ · · · +w₀s, x_st =1, x_t=1+s+t s+st s+ · · · +w₀t form a basis of the descent algebraΣ (W ). Note thatx_t+x_s=x_∅+1−w₀.

ForL⊆K⊆S, the numbersm_KL= |X_K∩X_L|are easily determined as

(mKL)K,L⊆S=

⎡

⎢⎣

2m . . .

m 2 . . m . 2 .

1 1 1 1

⎤

⎥⎦, (mKL)⁻¹=

⎡

⎢⎢

⎢⎣

1

2m . . .

−¹₄ ¹₂ . .

−¹₄ . ¹₂ .

m−1

2m −¹₂ −¹₂ 1

⎤

⎥⎥

⎥⎦.

Hence the idempotentse_Lare (cf. [2]) e_∅= 1

2mx_∅, e_s=1

2x_s−1 4x_∅, e_st =1−1

2x_s−1

2x_t+m−1

2m x_∅, e_t=1 2x_t−1

4x_∅.

Fromx_t+x_s =1+x_∅−w₀, it follows thate_s+e_t =¹₂(1−w₀), and hence that eS=¹₂(1+w0)−e_∅=Av(w0)−Av(W ), as required.

As an immediate consequence we obtain the character of the top component of CW.

Corollary 5.5 TheW-moduleE_[S]affords the characterΦS=Ind^W_w

0(1)−1S. Next we identify linear characters of centralizers of cuspidal elements. Note that the groupWconsists ofmreflections andmrotations. Denote the rotation subgroup of W by W⁺= st. The centralizer of a rotation w is W⁺, unless w is central inW. The cuspidal classes ofW are exactly the classes of nontrivial rotations, rep- resented by the set{(st )^j :j =1, . . . ,^m₂}, containingw0=(st )^m/2 in case mis even. The groupW⁺is a cyclic group of ordermand it hasmlinear charactersχj, j=0, . . . , m−1, defined by

χj(st )=ζ_m^j

for a primitivemth root of unityζm. In the following arguments, we make frequent use of the fact that the sum of all the nontrivial charactersχj of W⁺ equals the difference of its regular and its trivial character,

m−1 j=1

χj=Ind^W_{1}⁺(1)−1_W+,

which obviously follows from ^m_j=⁻₀¹χ_j=Ind^W_{1}⁺(1)andχ₀=1_W+. We distinguish two cases, depending on the parity ofm.

Proposition 5.6 Suppose thatm=2kwithk >0. Let ϕ_{(st )}j=

χ_2j, 0< j < k, _S, j=k.

Thenϕ_{(st )}j is a linear character ofC_W((st )^j), forj=1, . . . , k, and k

j=1

Ind^W_C

W((st )^j)(ϕ_{(st )}j)=_S+

k−1

j=1

Ind^W_W+(χ_2j)=Φ_S.

Proof Note thatC_W((st )^j)=W⁺andw₀lies in the kernel of the charactersϕ_{(st )}j= χ_2j, for allj=1, . . . , k−1. Hence theχ_2j can be regarded as a full set of nontrivial irreducible characters of the quotient groupW⁺/w0, whence their sum ^k_j⁻₌¹₁χ2j

equals the difference of its regular and its trivial characters. Thus, as a character of W⁺, we have

k−1

j=1

χ2j=Ind^W_w⁺

0(1)−1_W+. Therefore

_S+Ind^W_{W+ k−1}

j=1

χ_2j

=_S+Ind^W_w0(1)−Ind^W_W+(1)=Ind^W_w0(1)−1S=Φ_S,

where the penultimate equality holds because Ind^W_W+(1)=1S+_S. Proposition 5.7 Suppose thatm=2k+1 for somek >0. Forj=1, . . . , k, let

ϕ_{(st )}j =χ_j.

Thenϕ_{(st )}j is a linear character ofC_W((st )^j), forj=1, . . . , k, and k

j=1

Ind^W_C

W((st )^j)(ϕ_{(st )}j)= k j=1

Ind^W_W+(χ_j)=Φ_S.

Proof We haveC_W((st )^j)=W⁺and Res^W_W+(Ind^W_W+(χ_j))=χ_j+χ_m−j for allj= 1, . . . , k. Hence

Res^W_W+

_k

j=1

Ind^W_W+(χ_j)

=

m−1 j=1

χ_j=Ind^W_{1}⁺(1)−1W⁺

=Res^W_W+ Ind^W_w

0(1)−1S

=Res^W_W+(ΦS).

It follows that

Φ_S= k j=1

Ind^W_W+(χ_j),

since the restrictions of both characters to the subgroupw0ofWalso coincide.