Generalized descent algebras

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Preprint submitted on 7 Jan 2005

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Generalized descent algebras

Cédric Bonnafé, Christophe Hohlweg

To cite this version:

Cédric Bonnafé, Christophe Hohlweg. Generalized descent algebras. 2005. �hal-00003811�

ccsd-00003811, version 1 - 7 Jan 2005

C´EDRIC BONNAF´E AND CHRISTOPHE HOHLWEG¹

Abstract. IfAis a subset of the set of reflections of a finite Coxeter group W, we define a sub-Z-moduleD^A(W) of the group algebraZW. We provide examples where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra and, if W if of type B, the Mantaci-Reutenauer algebra.

Introduction

Let (W, S) be a finite Coxeter system whose length function is denoted by ℓ : W → N = {0, 1, 2, . . . }. In 1976, Solomon introduced a remarkable subalgebra ΣW of the group algebra ZW , called the Solomon descent algebra [8]. Let us recall its definition. If I ⊂ S, let W

I

denote the standard parabolic subgroup generated by I.

Then

X

I

= {w ∈ W | ∀ s ∈ I, ℓ(ws) > ℓ(w)}

is a set of minimal length coset representatives of W/W

I

. Let x

I

= P

w∈XI

w ∈ ZW . Then ΣW is defined as the sub- Z -module of Z W spanned by (x

I

)

I⊂S

. More- over, ΣW is endowed with a Z -linear map θ : ΣW → Z Irr W satisfying θ(x

I

) = Ind

^W_WI

1

WI

. This is an algebra homomorphism.

If W is the symmetric group S

_n

, θ becomes an epimorphism and the pair (Σ S

_n

, θ) provides a nice construction of Irr( S

_n

) [6], which is the first ingredient of several recent works, see for instance [9, 3]. However, the morphism θ is surjective if and only if W is a product of symmetric groups.

In [4], we have constructed a subalgebra Σ

^′

(W

n

) of the group algebra ZW

n

of the Coxeter group W

n

of type B

n

: it turns out that this subalgebra contains ΣW

n

, that it is also endowed with an algebra homomorphism θ

^′

: Σ

^′

(W

n

) → Z Irr W

n

extending θ. Moreover, θ

^′

is surjective and Q ⊗

Z

Ker θ

^′

is the radical of Q ⊗

Z

Σ

^′

(W

n

). This lead to a construction of the irreducible characters of W

n

following J˝ollenbeck’s strategy. In fact, Σ

^′

(W

n

) is the Mantaci-Reutenauer algebra [7].

It is a natural question to ask whether this kind of construction can be generalized to other groups. However, the situation seems to be much more complicated in the other types. Let us explain now what kind of subalgebras we are looking for.

Let T = {wsw

⁻¹

| w ∈ W and s ∈ S} be the set of reflections in W . Let A be a fixed subset of T . If I ⊂ A, we still denote by W

I

the subgroup of W generated by I and we still set

X

I

= {w ∈ W | ∀ x ∈ W

I

, ℓ(wx) ≥ ℓ(w)}.

Date: January 7, 2005.

1Partially supported by Canada Research Chairs 1

Then X

I

is again a set of representatives for W/W

I

. Now, let x

I

= P

w∈XI

w ∈ ZW . Then

Σ

A

(W ) = X

I⊂A

Z x

I

is a sub-Z-module of ZW . However, it is not in general a subalgebra of ZW . Let us define another sub-Z-module of ZW . If w ∈ W , let

D

A

(w) = {s ∈ A | ℓ(ws) < ℓ(w)} ⊂ A

be the A-descent set of w. A subset I of A is said to be A-admissible if there exists w ∈ W such that D

A

(w) = I. Let P

ad

(A) denote the set of A-admissible subsets of A. If I ∈ P

ad

(A), we set D

^A_I

= {w ∈ W | D

A

(w) = I} and d

^A_I

= P

w∈DI^A

w ∈ ZW . Now, let

D

A

(W ) = ⊕

I∈Pad(A)

Z d

^A_I

.

As an example, D

S

(W ) = Σ

S

(W ) = ΣW . The main theorem of this paper is the following (here, C(w) denotes the conjugacy class of w in W ):

Theorem A. If there exists two subsets S

1

and S

2

of S such that A = S

1

∪ (∪

s∈S2

C(s)), then D

A

(W ) is a subalgebra of Z W .

This theorem contains the case of Solomon descent algebra (take A = S), the Mantaci-Reutenauer algebra Σ

^′

(W

n

) (take A = {s

1

, . . . , s

n−1

} ∪ C(t), where S = {t, s

1

, . . . , s

n−1

} satisfies C(t) ∩ {s

1

, . . . , s

n−1

} = ∅ ), and one of the subalgebras we have constructed in type G

2

. But it also gives a new algebra in type F

4

of Z -rank 300 (take A = {s

1

, s

2

} ∪ C(s

3

), where S = {s

1

, s

2

, s

3

, s

4

} satisfies {s

1

, s

2

} ∩ C(s

3

) = ∅ ).

Moreover, if A = T , we get that D

A

(W ) = ZW (see Example 1.7). In the case of dihedral groups, we get another family of algebras:

Theorem B. If W is a dihedral group of order 4m (m ≥ 1), S = {s, t} and A = {s, t, sts} or A = {t, sts}, then D

A

(W ) is a subalgebra of ZW .

It must be noted that the algebras constructed in Theorems A and B are not necessarily unitary. More precisely 1 ∈ D

A

(W ) if and only if S ⊂ A. Moreover, if S ⊂ A, then ΣW ⊂ D

A

(W ).

The bad point in the above construction is that, in many of the above cases, one has Σ

A

(W ) 6= D

A

(W ). Also, we are not able to construct in general a morphism of algebras θ

A

: D

A

(W ) → Z Irr W extending θ if S ⊂ A. It can be proved that, in some cases, this morphism does not exist.

This paper is organized as follows. Section 1 is essentially devoted to the proofs of Theorems A and B. One of the key step is that the subsets D

^A_I

are left-connected (recall that a subset E of W is said to be left-connected if, for all w, w

^′

∈ E, there exists a sequence w = w

1

, w

2

,. . . , w

r

= w

^′

of elements of E such that w

i+1

w

⁻¹_i

∈ S for every i ∈ {1, 2, . . . , r − 1}). In Section 2, we discuss more precisely the case of dihedral groups.

1. Descent sets

Let (W, S) be a finitely generated Coxeter system (not necessary finite). If s, s

^′

∈ S , we denote by m(s, s

^′

) the order of ss

^′

∈ W . If W is finite, we denote by w

0

its longest element.

1.1. Root system. Let V be an R-vector space endowed with a basis indexed by S denoted by ∆ = {α

s

| s ∈ S }. Let B : V × V → R be the symmetric bilinear form such that

B(α

s

, α

s^′

) = − cos π

m(s, s

^′

)

for all s, s

^′

∈ S. If s ∈ S and v ∈ V , we set

s(v) = v − 2B(α

s

, v)α

s

.

Thus s acts as the reflection in the hyperplane orthogonal to α

s

(for the bilinear form B). This extends to an action of W on V as a group generated by reflections.

It stabilizes B.

We recall some basic terminology on root systems. The root system of (W, S) is the set Φ = {w(α

s

) | w ∈ W, s ∈ S} and the elements of ∆ are the simple roots.

The roots contained in

Φ

⁺

= X

α∈∆

R

⁺

α

∩ Φ

are said to be positive, while those contained in Φ

⁻

= −Φ

⁺

are said to be negative.

Moreover, Φ is the disjoint union of Φ

⁺

and Φ

⁻

. If w ∈ W , ℓ(w) = |N (w)| , where N (w) =

α ∈ Φ

⁺

| w(α) ∈ Φ

⁻

.

Let α = w(α

s

) ∈ Φ, then s

α

= wsw

⁻¹

acts as the reflection in the hyperplane orthogonal to α and s

α

= s

−α

. Therefore, the set of reflections of W

T = [

w∈W

wSw

⁻¹

is in bijection with Φ

⁺

(and thus Φ

⁻

).

Let us recall the following well-known result:

Lemma 1.1. Let w ∈ W . Then:

(a) If α ∈ Φ

⁺

, then ℓ(ws

α

) > ℓ(w) if and only if w(α) ∈ Φ

⁺

. (b) If s ∈ S, then

N (sw) =

( N (w) ` {w

⁻¹

(α

s

)} if ℓ(sw) > ℓ(w), N (w) \ {−w

⁻¹

(α

s

)} otherwise.

From now on, and until the end of this paper, we fix a subset A of T . We start with easy observations.

As a consequence of Lemma 1.1 (a), we get that D

A

(w) =

s

α

∈ A | α ∈ Φ

⁺

and w(α) ∈ Φ

⁻

. We also set

N

A

(w) = {α ∈ Φ

⁺

| s

α

∈ A and w(α) ∈ Φ

⁻

}.

The map N

A

(w) → D

A

(w), α 7→ s

α

is then a bijection.

1.2. Properties of the map D

_A

. First, using Lemma 1.1 (b), we get:

Corollary 1.2. If s ∈ S and if w ∈ W is such that w

⁻¹

sw = s

_w−1(αs)

6∈ A, then N

A

(w) = N

A

(sw) (and D

A

(w) = D

A

(sw)).

Remark 1.3 - If A

1

⊂ A

2

⊂ T , then D

A1

(w) = D

A2

(w) ∩ A

1

for all w ∈ W . Therefore if W is finite, D

A1

(W ) ⊂ D

A2

(W ).

Proposition 1.4. We have:

(a) ∅ is A-admissible.

(b) D

∅^A

= {1} if and only if S ⊂ A.

Proof. We have D

A

(1) = ∅ so (a) follows. If s ∈ S \ A, then D

A

(s) = ∅. This

shows (c).

The notion of A-descent set is obviously compatible with direct products:

Proposition 1.5. Assume that W = W

1

× W

2

where W

1

and W

2

are standard parabolic subgroups of W . Then, for all I ∈ P

ad

(A), we have

D

_I^A

= D

^A∩W_I∩W1¹

× D

^A∩W_I∩W2²

.

Corollary 1.6. Assume that W is finite and that W = W

1

× W

2

where W

1

and W

2

are standard parabolic subgroups of W . Then

D

A

(W ) = D

A∩W1

(W

1

) ⊗

Z

D

A∩W2

(W

2

).

Example 1.7 - Consider the case where A = T (then N

A

(w) = N(w)). It is well- known [5, Chapter VI, Exercise 16] that the map w 7→ N (w) from W onto the set of subsets of Φ

⁺

is injective (observe that if α ∈ N(w

1

w

⁻¹₂

) then ±w

⁻¹₂

(α) lives in the union, but not in the intersection, of N(w

1

) and N (w

2

)). Therefore, the map W → P

ad

(T ), w 7→ D

T

(w) is injective. In particular, if W is finite, then D

T

(W ) = ZW .

In the case of finite Coxeter groups, the multiplication on the left by the longest element has the following easy property.

Proposition 1.8. If W is finite and if w ∈ W , then D

A

(w

0

w) = A \ D

A

(w).

Corollary 1.9. If W is finite, then:

(a) A is A-admissible;

(b) I ∈ P

ad

(A) if and only if A \ I ∈ P

ad

(A);

(c) D

_A^A

= {w

0

} if and only if S ⊂ A.

Proof. D

A

(w

0

) = A so (a) follows. (b) follows from Proposition 1.8. (c) follows

from Proposition 1.8 and Proposition 1.4 (b).

1.3. Left-connectedness. In [1], Atkinson gave a new proof of Solomon’s result by using an equivalence relation to describe descent sets. We extend his result to A-descent sets. It shows in particular that the subsets D

^A_I

are left-connected.

Let w and w

^′

be two elements of W . We say that w is an A-descent neighborhood of w

^′

, and we write w ⌣

A

w

^′

, if w

^′

w

⁻¹

∈ S and w

⁻¹

w

^′

∈ / A. It is easily seen that

⌣

A

is a symmetric relation. The reflexive and transitive closure of the A-descent

neighborhood relation is called the A-descent equivalence, and is denoted by ∼

A

.

The next proposition characterizes this equivalence relation in terms of A-descent

sets.

Proposition 1.10. Let w, w

^′

∈ W . Then

w ∼

A

w

^′

⇔ D

A

(w) = D

A

(w

^′

) ⇔ N

A

(w) = N

A

(w

^′

).

Proof. The second equivalence is clear. If w ⌣

A

w

^′

, then it follows from Corollary 1.2 that N

A

(w) = N

A

(w

^′

). It remains to show that, if N

A

(w) = N

A

(w

^′

), then w ∼

A

w

^′

.

So, assume that N

A

(w) = N

A

(w

^′

). Write x = w

^′

w

⁻¹

and let m = ℓ(x). If ℓ(x) = 0, then w = w

^′

and we are done. Assume that m ≥ 1 and write x = s

1

s

2

. . . s

m

with s

i

∈ S. We now want to prove on induction on m that

(∗) w ⌣

A

s

m

w ⌣

A

s

m−1

s

m

w ⌣

A

· · · ⌣

A

s

2

. . . s

m

w ⌣

A

s

1

s

2

. . . s

m

w = w

^′

. First, assume that w 6⌣

A

s

m

w. In other words, w

⁻¹

s

m

w ∈ A. For simplification, let α

i

= α

sⁱ

. By Lemma 1.1 (b), we have

N

A

(s

m

w) = N

A

(w) a

{w

⁻¹

(α

m

)} or N

A

(w) \ {−w

⁻¹

(α

m

)}.

In the first case, as N

A

(w) = N

A

(w

^′

), and by applying again Lemma 1.1 (b), there exists a step i ∈ {1, 2, . . . , m − 1} between N

A

(w) to N

A

(w

^′

) where w

⁻¹

(α

m

) is removed from N

A

(s

i

. . . s

m

w), that is, (s

i+1

. . . s

m

w)

⁻¹

(α

i

) = −w

⁻¹

(α

m

). In the same way, we get the same result in the second case. In other words, we have proved that there exists i ∈ {1, 2, . . . , m − 1} such that s

m

. . . s

i+1

(α

i

) = −α

m

, so, by Lemma 1.1 (a), we have ℓ(s

m

. . . s

i+1

s

i

) < ℓ(s

m

. . . s

i+1

). This contradicts the fact that m = ℓ(x). So w ⌣

A

s

m

w, and then N

A

(w) = N

A

(s

m

w). Hence N

A

(s

m

w) = N

A

(w

^′

) and w

^′

(s

m

w)

⁻¹

= s

1

s

2

. . . s

m−1

. We get then by induction that s

m

w ⌣

A

s

m−1

s

m

w ⌣

A

· · · ⌣

A

s

1

s

2

. . . s

m

w = w

^′

, which shows (∗).

Corollary 1.11. If I ∈ P

ad

(A), then D

_I^A

is left-connected.

1.4. Nice subsets of T . We say that A is nice if, for every s ∈ A and w ∈ W such that w

⁻¹

sw 6∈ A, we have D

A

(sw) = D

A

(w). Notice that every subset of S is nice, by Corollary 1.2.

If w ∈ W and I, J ∈ P

ad

(A), we set D

A

(I, J, w) =

(u, v) ∈ D

^A_I

× D

^A_J

| uv = w .

The next lemma gives an obvious characterization of the fact that D

A

(W ) is an algebra in terms of these sets.

Lemma 1.12. Assume that W is finite. Then the following are equivalent:

(1) D

A

(W ) is a subalgebra of ZW .

(2) For all I, J ∈ P

ad

(A) and for all w, w

^′

∈ W such that D

A

(w) = D

A

(w

^′

), we have |D

A

(I, J, w)| = |D

A

(I, J, w

^′

)|.

If these conditions are fulfilled, we choose for any I ∈ P

ad

(A) an element z

I

in D

_I^A

. Then

d

^A_I

d

^A_J

= X

K∈Pad(A)

|D

A

(I, J, z

K

)|d

^A_K

.

Let us fix now s ∈ S and let (u, v) ∈ W × W . If u ⌣

A

su, we set ψ

_s^A

(u, v) =

(su, v). If u 6⌣

A

su, then we set ψ

_s^A

(u, v) = (u, u

⁻¹

suv). Note that, in the last

case, u

⁻¹

su ∈ A. We have (ψ

_s^A

)

²

= Id

W×W

. In particular, ψ

_s^A

is a bijection. Using

ψ

^A_s

, one can relate the notion of nice subsets to the property (2) stated in Lemma

1.12.

Proposition 1.13. Assume that A is nice. Let I, J ∈ P

ad

(W ), let w ∈ W and let s ∈ S be such that w ⌣

A

sw. Then ψ

^A_s

(D

A

(I, J, w)) = D

A

(I, J, sw).

Proof. Let (u, v) be an element of D

A

(I, J, w). By symmetry, we only need to prove that ψ

^A_s

(u, v) ∈ D

A

(I, J, sw). If u ⌣

A

su, then D

A

(su) = D

A

(u) = I by Proposition 1.10, so ψ

_s^A

(u, v) = (su, v) ∈ D

A

(I, J, sw). So, we may, and we will, assume that u 6⌣

A

su. Let s

^′

= u

⁻¹

su ∈ A. Note that w

⁻¹

sw = v

⁻¹

s

^′

v 6∈ A.

Then, ψ

_s^A

(u, v) = (u, s

^′

v) and us

^′

v = sw. So, we only need to prove that D

A

(s

^′

v) = D

A

(v). But this just follows from the definition of nice subset of T . Corollary 1.14. If W is finite and if A is nice, then D

A

(W ) is a subalgebra of ZW . It is unitary if and only if S ⊂ A.

Proof. This follows from Lemma 1.12 and from Propositions 1.10 and 1.13.

Remark 1.15 - It would be interesting to know if the converse of Corollary 1.14 is true.

1.5. Proof of Theorems A and B. Using Corollary 1.14, we see that Theorems A and B are direct consequences of the following theorem (which holds also for infinite Coxeter groups):

Theorem 1.16. Assume that one of the following holds:

(1) There exists two subsets S

1

and S

2

of S such that A = S

1

∪ (∪

s∈S2

C(s)).

(2) S = {s, t}, m(s, t) is even or ∞, and A = {s, t, sts} or A = {t, sts}.

Then A is nice.

Proof. Assume that (1) or (2) holds. Let r ∈ A and let w ∈ W be such that w

⁻¹

rw 6∈ A. We want to prove that D

A

(rw) = D

A

(w). By symmetry, we only need to show that D

A

(rw) ⊂ D

A

(w). If r ∈ S, then this follows from Corollary 1.2. So we may, and we will, assume that r / ∈ S.

• Assume that (1) holds. Write A

^′

= ∪

s∈S2

C(s) and S

^′

= A \ A

^′

. Then S

^′

⊂ S, A = A

^′

` S

^′

and A

^′

is stable under conjugacy. Then r ∈ A

^′

and w

⁻¹

rw ∈ A

^′

⊂ A, which contradicts our hypothesis.

• Assume that (2) holds. If m(s, t) = 2, then A is contained in S and therefore is nice by Corollary 1.2. So we may, and we will, assume that m(s, t) ≥ 4. Since r 6∈ S , we have r = sts. Assume that D

A

(rw) 6⊂ D

A

(w). Let

Φ

A

= {α ∈ Φ

⁺

| s

α

∈ A} ⊂ {α

s

, α

t

, s(α

t

)}.

There exists α ∈ Φ

A

such that rw(α) ∈ Φ

⁻

and w(α) ∈ Φ

⁺

. So w(α) ∈ N(r) = {α

s

, s(α

t

), st(α

s

)}. Since w

⁻¹

rw 6∈ A, we have that ws

α

w

⁻¹

6= r, so w(α) 6= s(α

t

).

So w(α) = α

s

or st(α

s

). But the roots α

s

and α

t

lie in different W -orbits. So α = α

s

and w(α

s

) ∈ {α

s

, st(α

s

)}. If W is infinite, this gives that w ∈ {1, st}.

This contradicts the fact that w

⁻¹

rw 6∈ A. If W is finite, this gives that w ∈ {1, st, w

0

s, stsw

0

}. But again, this contradicts the fact that w

⁻¹

rw 6∈ A.

Example 1.17 - Assume here that W is of type F

4

, that S = {s

1

, s

2

, s

3

, s

4

} and that A = C(s

1

) ∪ S. Then, using GAP , one can see that rank

Z

D

A

(W ) = 300 and rank

Z

Σ

A

(W ) = 149. Moreover, Σ

A

(W ) is not a subalgebra of D

A

(W ).

Some computations with GAP suggest that the following question has a positive

answer:

Question. Let A be a nice subset of T containing S. Is it true that A is one of the subsets mentioned in Theorem 1.16?

2. The dihedral groups

The aim of this section is to study the unitary subalgebras D

A

(W ) constructed in Theorems A and B whenever W is finite and dihedral.

Hypothesis: From now on, and until the end of this paper, we assume that S = {s, t} with s 6= t and that m(s, t) = 2m, with 2 ≤ m < ∞.

Note that w

0

= (st)

^m

is central. We denote by P

0

(A) the set of subsets I of A such that W

I

∩ A = I. With this notation, we have

Σ

A

(W ) = X

I∈P0(A)

Z x

I

.

We now define an equivalence relation ≡ on P

0

(A): if I, J ∈ P

0

(A), we write I ≡ J if W

I

and W

J

are conjugate in W . We set

Σ

⁽¹⁾_A

(W ) = X

I,J∈P0(A) I≡J

Z (x

I

− x

J

).

In what follows, we will also need some facts on the character table of W . Let us recall here the construction of Irr W . First, let H be the subgroup of W generated by st. It is normal in W , of order 2m (in other words, of index 2). We choose the primitive (2m)-th root of unity ζ ∈ C of argument

_m^π

. If i ∈ Z, we denote by ξ

i

: H → C

^×

the unique linear character such that ξ

i

(st) = ζ

ⁱ

. Then Irr H = {ξ

i

| 0 ≤ i ≤ 2m − 1}. Now, let

χ

i

= Ind

^W_H

ξ

i

.

Then χ

i

= χ

2m−i

and, if 1 ≤ i ≤ m − 1, χ

i

∈ Irr W . Also, χ

i

has values in R. More precisely, for 1 ≤ i ≤ m − 1 and j ∈ Z

χ

i

(ts)

^j

= ζ

^j

+ ζ

^−j

= 2 cos ijπ

m

and χ

i

s(ts)

^j

= 0.

Let 1 denote the trivial character of W , let ε denote the sign character and let γ : W → {1, −1} be the unique linear character such that γ(s) = −γ(t) = 1. Then (2.1) Irr W = {1, ε, γ, εγ} ∪ {χ

i

| 1 ≤ i ≤ m − 1}.

In particular, | Irr W | = m + 3.

2.1. The subset A = {s, t, sts}. From now on, we assume that A = {s, t, sts}.

We set ¯ s = A \ {s} = {t, sts} and ¯ t = A \ {t} = {s, sts}. It is easy to see that P

ad

(A) = {∅, {s}, {t}, ¯ s, t, A}. ¯

For simplification, we will denote by d

I

the element d

^A_I

of Z W (for I ∈ P

ad

(W )) and we set d

s

= d

{s}

and d

t

= d

{t}

. We have

d

∅

= 1, d

s¯

= w

0

s,

d

s

= s, d

A

= w

0

,

d

t

=

m−1

X

i=1

(st)

ⁱ

+ (ts)

ⁱ⁻¹

t

, d

¯t

=

m−1

X

i=1

(st)

ⁱ

s + (ts)

ⁱ

.

The multiplication table of D

A

(W ) is given by

1 d

_s

d

_¯s

d

_A

d

_t

d

¯t

1 1 d

s

d

_¯s

d

A

d

t

d

¯t

d

s

d

s

1 d

A

d

s_¯

d

t

d

¯t

d

s¯

d

¯s

d

A

1 d

s

d

¯t

d

t

d

A

d

A

d

¯s

d

s

1 d

¯t

d

t

d

t

d

t

d

¯t

d

t

d

¯t

z

A

z

A

d

¯t

d

¯t

d

_t

d

t¯

d

_t

z

_A

z

_A

where z

A

= (m − 1) 1 + d

A

+ d

s

+ d

s¯

+ (m − 2) d

t

+ d

¯t

. We now study the sub- Z-module Σ

A

(W ): we will show that it coincides with D

A

(W ). First, it is easily seen that

P

0

(A) = {∅, {s}, {t}, {sts}, ¯ s, A}

and that

x

A

= 1 x

s¯

= 1 +d

s

x

sts

= 1 +d

s

+d

t

x

t

= 1 +d

s

+d

¯t

x

s

= 1 +d

t

+d

¯s

x

∅

= 1 +d

s

+d

t

+d

¯t

+d

¯s

+d

A

Therefore, Σ

A

(W ) = D

A

(W ) = ⊕

I∈P0(A)

Zx

I

. So we can define a map θ

A

: Σ

A

(W ) → Z Irr W by θ

A

(x

I

) = Ind

^W_WI

1

W^I

.

Proposition 2.2. Assume that S = {s, t} with s 6= t, that m(s, t) = 2m with m ≥ 2 and that A = {s, t, sts}. Then:

(a) Σ

A

(W ) = D

A

(W ) is a subalgebra of Z W of Z -rank 6.

(b) θ

A

is a morphism of algebras.

(c) Ker θ

A

= Z (x

t

− x

sts

) = Σ

⁽¹⁾_A

(W ).

(d) Q ⊗

Z

Ker θ

A

is the radical of Q ⊗

Z

Σ

A

(W ).

(e) θ

A

is surjective if and only if m = 2 that is, if and only if W is of type B

2

. Proof. (a) has already been proved. For proving the other assertions, we need to compute explicitly the map θ

A

. It is given by the following table:

d

I

1 d

s

d

s¯

d

A

d

t

d

t¯

θ

A

(d

I

) 1 εγ γ ε

m−1

X

i=1

χ

i m−1

X

i=1

χ

i

(c) This shows that Ker θ

A

= Z(d

¯t

− d

t

) = Z(x

t

− x

sts

) = Σ

⁽¹⁾_A

(W ), so (c) holds.

(b) To prove that θ

A

is a morphism of algebras, the only difficult point is to prove that θ

A

(d

²_t

) = θ

A

(d

t

)

²

. Let ρ denote the regular character of W . Then

θ

A

(d

t

) = 1

2 ρ − 1 − γ − εγ − ε).

Therefore,

θ

A

(d

t

)

²

= (m − 2)ρ + 1 + γ + εγ + ε.

But, d

²_t

= z

A

= (m − 2)x

∅

+ 1 + d

s

+ d

s¯

+ d

A

. This shows that θ

A

(d

²_t

) = θ

A

(d

t

)

²

. (d) Let R denote the radical of Q ⊗

Z

Σ

A

(W ). We only need to prove that C ⊗

Q

R = C ⊗

Z

Σ

A

(W ). Since C Irr W is a split semisimple commutative algebra, every subalgebra of C Irr W is semisimple. So ( C ⊗

Z

Σ

A

(W ))/( C ⊗

Z

Ker θ

A

) is a semisimple algebra. This shows that R is contained in C ⊗

Z

Ker θ

A

. Moreover, since (x

t

− s

sts

)

²

= (d

t

− d

¯t

)

²

= 0, Ker θ

A

is a nilpotent two-sided ideal of Σ

A

(W ).

So C ⊗

Z

Ker θ

A

is contained in C ⊗

Z

R. This shows (d).

(e) If m = 2, then Irr W = {1, γ, εγ, ε, χ

1

} = θ

A

({1, d

s

, d

s¯

, d

A

, d

t

}) so θ

A

is surjective. Conversely, if θ

A

is surjective, then | Irr W | = 5 (by (a) and (c)). Since

| Irr W | = m + 3, this gives m = 2.

We close this subsection by giving, for Σ

A

(W ), a complete set of orthogonal prim- itive idempotents, extending to our case those given in [2], and the corresponding irreducible representations:

E

∅

= 1

4m x

∅

, E

s

= 1 2

x

s

− 1

2 x

∅

, E

t

= 1 2

x

t

− 1

2 x

∅

,

E

s¯

= 1 2

x

s¯

− 1

2 x

t

− 1

2 x

sts

+ m − 1 2m x

∅

E

A

= 1 − 1 2 x

s

− 1

4 x

t

+ 1

4 x

sts

− 1 2 x

s¯

+ 1

4 x

∅

For i ∈ { ∅ , s, t, s, A}, we denote by ¯ P

i

the indecomposable projective module Q Σ

A

(W )E

i

. Let Q

^A_i

denote the unique simple Q Σ

A

(W )-module lying in the head of P

i

. It is easily seen, using the previous multiplication table, that

dim P

∅

= dim P

t

= dim P

¯s

= dim P

A

= 1

and that dim P

s

= 2

(note that x

¯s

E

s

= −(x

t

− x

sts

)). In other words,

P

∅

= Q

^A_∅

, P

t

= Q

^A_t

, P

s¯

= Q

^A_s¯

, P

A

= Q

^A_A

and Rad(P

s

) ≃ Q

^A_t

.

For w ∈ W , we denote by ev

w

the morphism of algebras Z Irr W → Z, χ 7→ χ(w) and we set ev

^A_w

= ev

w

◦ θ

A

. Then the morphism of algebras Σ

A

(W ) → Z associated to the simple module Q

^A_i

is ev

^A_f(i)

, where

f ( ∅ ) = 1, f (s) = s, f (t) = t, f (¯ s) = (st)

^m

= w

0

and f (A) = st.

2.2. The subset B = {s} ∪ C (t). Let B = {s} ∪ C(t) (so that |B| = m + 1). It is easy to see that P

ad

(B) consists of the sets

∅ = D

B

(1), B = D

B

(w

0

), {s} = D

B

(s), C(t) = D

B

(w

0

s), D

B

(ts)

ⁱ

= D

B

s(ts)

ⁱ

, 1 ≤ i ≤ m − 1, D

B

(st)

^j

= D

B

(ts)

^j−1

t

, 1 ≤ j ≤ m − 1.

Therefore D

B

(W ) is a subalgebra of ZW of Z-rank (2m + 2).

Using GAP , we can see that, in general, Σ

B

(W ) 6= D

B

(W ). First examples are given in the following table

m 2 3 4 5 6 7 8 9 10 11

Z-rank of D

B

(W ) 6 8 10 12 14 16 18 20 22 24 Z -rank of Σ

B

(W ) 6 8 10 10 14 12 18 18 22 16

Remark 2.3 - The linear map θ

B

: Σ

B

(W ) → Z Irr W , x

I

7→ Ind

^W_WI

1

WI

, is well- defined and surjective if and only if m ∈ {2, 3} (recall that m ≥ 2). Indeed, the image of θ

B

can not contain a non-rational character. But all characters of W are rational if and only if W is a Weyl group, that is, if and only if 2m ∈ {2, 3, 4, 6}.

Moreover, if m = 2, then A = B and θ

B

= θ

A

is surjective by Proposition 2.2.

If m = 3, then this follows from Proposition 2.4 below.

2.3. The algebra D

B

(G

₂

). From now on, and until the end, we assume that m = 3. That is W is of type I

2

(6) = G

2

. For convenience, we keep the same notation as in §2.1. We have

d

∅

= 1, d

s

= s,

d

¯s

= d

^B_C(t)

= w

0

s, d

A

= d

^B_B

= w

0

,

d

1

= d

^B_{t}

= t + st, d

2

= d

^B_{s,sts}

= ts + sts,

d

3

= d

^B{s,sts,tstst}

= tsts + ststs, d

4

= d

^B_{t,tstst}

= tst + stst.

Let us now show that Σ

B

(W ) = D

B

(W ). First, it is easily seen that P

0

(B) = {∅, {s}, {t}, {sts}, s, ¯ {tstst}, {s, tstst}, B}

and that

x

A

= x

B

= 1 x

¯s

= 1 +d

s

x

{s,tstst}

= 1 +d

1

x

tstst

= 1 +d

s

+d

1

+d

2

x

t

= 1 +d

s

+d

2

+d

3

x

sts

= 1 +d

s

+d

1

+d

4

x

s

= 1 +d

1

+d

4

+d

s¯

x

∅

= 1 +d

s

+d

1

+d

2

+d

3

+d

4

+d

s¯

+d

A

Therefore, Σ

B

(W ) = D

B

(W ) = ⊕

I∈P0(B)

Zx

I

. So we can define a map θ

B

: Σ

B

(W ) → Z Irr W by θ

B

(x

I

) = Ind

^W_WI

1

W^I

.

Proposition 2.4. Assume that S = {s, t} with s 6= t, that m(s, t) = 6 and that B = {s, t, sts, tstst}. Then:

(a) Σ

B

(W ) = D

B

(W ) is a subalgebra of ZW of Z-rank 8.

(b) θ

B

is a surjective linear map (and not a morphism of algebras).

(c) Ker θ

B

= Z(x

tstst

− x

t

) ⊕ Z(x

tstst

− x

sts

) = Σ

⁽¹⁾_B

(W ).

(d) Irr W = θ

B

({1, d

s

, d

s¯

, d

A

, d

1

, d

2

}).

Proof. The map θ

B

is given by the following table:

d

^B_I

1 d

s

d

s¯

d

A

d

1

d

2

d

3

d

4

θ

B

(d

^B_I

) 1 εγ γ ε χ

2

χ

1

χ

2

χ

1

This shows (b) and (d). As d

1

− d

3

= x

{tstst}

− x

t

and d

2

− d

4

= x

tstst

− x

sts

, (c) is proved. Finally, the fact that θ

B

is not a morphism of algebras follows from θ

B

(d

²₁

)(w

0

) = (1 + εγ + χ

1

)(w

0

) = −2 6= θ

B

(d

1

)

²

(w

0

) = χ

2

(w

0

)

²

= 4.

For information, the multiplication table in the basis (d

^B_I

)

I∈Pad(B)

is given by

1 d

_s

d

₁

d

₂

d

₃

d

₄

d

_s¯

d

_A

1 1 d

s

d

1

d

2

d

3

d

4

d

s¯

d

A

d

s

d

s

1 d

1

d

2

d

3

d

4

d

A

d

s¯

d

₁

d

₁

d

₂

1 + d

s

+ d

₄

1 + d

s

+ d

₃

d

₂

+ d

s_¯

+ d

A

d

₁

+ d

_¯s

+ d

A

d

₄

d

₃

d

₂

d

₂

d

₁

1 + d

s

+ d

₄

1 + d

s

+ d

₃

d

₂

+ d

s_¯

+ d

A

d

₁

+ d

_¯s

+ d

A

d

₃

d

₄

d

₃

d

₃

d

₄

d

₂

+ d

_s¯

+ d

_A

d

₁

+ d

_s¯

+ d

_A

1 + d

_s

+ d

₄

1 + d

_s

+ d

₃

d

₂

d

₁

d

₄

d

₄

d

₃

d

₂

+ d

_s¯

+ d

_A

d

₁

+ d

_s¯

+ d

_A

1 + d

_s

+ d

₄

1 + d

_s

+ d

₃

d

₁

d

₂

d

_¯s

d

_s¯

d

_A

d

₃

d

₄

d

₁

d

₂

1 d

_s

d

A

d

A

d

s¯

d

3

d

4

d

1

d

2

d

s

1

References

[1] M. D. Atkinson: A new proof of a theorem of Solomon, Bull. London Math. Soc.18(1986), 351–354.

[2] F. Bergeron, N. Bergeron, R. B. Howlett and D. E. Taylor: A decomposition of the descent algebra of a finite Coxeter group, J. Algebraic Combin.1(1992), 23–44.

[3] D. Blessenohl, M. Schocker: Noncommutative Character Theory of Symmetric groups I, preprint (2002).

[4] C. Bonnaf´e, C. Hohlweg: Generalized descent algebra and construction of irreducible char- acters of hyperoctahedral groups, preprint (2004).

[5] N. Bourbaki: Groupes et alg`ebres de Lie, Chap. 4-6, Hermann (1968).

[6] A. J˝ollenbeck: Nichtkommutative Charaktertheorie der symmetrischen Gruppen, Bayreuth.

Math. Schr.56(1999), 1–41.

[7] R. Mantaci, C. Reutenauer: A generalization of Solomon’s algebra for hyperoctahedral groups and other wreath products, Comm. in Algebra23(1)(1995), 27–56.

[8] L. Solomon: A Mackey formula in the group ring of a Coxeter group, J. Algebra41(1976), 255–268.

[9] J.-Y. Thibon, Lectures on noncommutative symmetric functions, inInteraction of combi- natorics and representation theory,pp. 39–94, Math. Soc. Japan Memoirs, vol. 11, Tokyo:

Math. Soc. Japan (2001).

Universit´e de Franche-Comt´e, D´epartement de Math´ematiques, 16 route de Gray, 25 000 Besanc¸on, France

E-mail address: bonnafe@descartes.univ-fcomte.fr

The Fields Institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1 E-mail address: chohlweg@fields.utoronto.ca