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Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups
Cédric Bonnafé, Christophe Hohlweg
To cite this version:
Cédric Bonnafé, Christophe Hohlweg. Generalized descent algebra and construction of irreducible
characters of hyperoctahedral groups. Annales de l’Institut Fourier, Association des Annales de
l’Institut Fourier, 2008, 56, pp.131-181. �hal-00002827v2�
ccsd-00002827, version 2 - 14 Sep 2004
AND CONSTRUCTION OF IRREDUCIBLE CHARACTERS OF HYPEROCTAHEDRAL GROUPS
C´EDRIC BONNAF´E AND CHRISTOPHE HOHLWEG
1. Introduction
Let (W, S) be a finite Coxeter system and let ℓ : W → N denote the length function. If I ⊂ S, WI =< I > is the standard parabolic subgroup generated by I and XI = {w ∈ W | ∀ s ∈ I, ℓ(ws) > ℓ(w)} is a cross-section of W/WI. WritexI =P
w∈XIw∈ ZW, then Σ(W) =⊕I⊂SZxI is a subalgebra of ZW and the Z-linear mapθ : Σ(W) →ZIrrW, xI 7→ IndWWI1 is a morphism of algebras:
Σ(W) is called thedescent algebraor theSolomon algebraofW [18]. However, the morphismθis surjective if and only if W is a product of symmetric groups.
The aim of this paper is to construct, whenever W is of type C, a subalge- bra Σ′(W) of ZW containing Σ(W) and a surjective morphism of algebras θ′ : Σ′(W) → ZIrrW build similarly as Σ(W) by starting with a bigger generating set. More precisely, let (Wn, Sn) denote a Coxeter system of type Cn and write Sn={t, s1, . . . , sn−1} where the Dynkin diagram of (Wn, Sn) is
. . .
t s1 s2 sn−1
Let t1 = t and ti = si−1ti−1si−1 (2 ≤ i ≤ n) and Sn′ = Sn∪ {t1, . . . , tn}. Let P0(Sn′) denote the set of subsets I ofSn′ such thatI=< I >∩Sn′. IfI∈ P0(Sn′), letWI,XI andxI be defined as before. Then:
Theorem. Σ′(Wn) =⊕I∈P0(Sn′)ZxI is a subalgebra ofZWn and the Z-linear map θn : Σ′(Wn) → ZIrrWn, xI 7→ IndWWI1 is a surjective morphism of algebras.
Moreover, Kerθn = P
I≡I′Z(xI −xI′) and Q⊗ZKerθn is the radical of the Q- algebra Q⊗ZΣ′(Wn).
In this theorem, the notationI≡I′ means that there existsw∈Wn such that I′=wI, that is,WI andWI′ are conjugated. This theorem is stated and proved in
§3.3. Note that it is slightly differently formulated: in fact, it turns out that there is a natural bijection between signed compositions ofnandP0(Sn′) (see Lemma 2.5).
So, everything in the text is indexed by signed compositions instead of P0(S′n). It must also be noticed that, by opposition with the classical case, the multiplication xIxJ may involve negative coefficients. Using another basis, we show that Σ′(Wn) is precisely the generalized descent algebra discovered by Mantaci and Reutenauer [16].
Using this theorem and the Robinson-Schensted correspondence for typeC con- structed by Stanley [20] and a Knuth version of it given in [5], we obtain an analog
1
of J¨ollenbeck’s result (on the construction of characters of the symmetric group [12]) using an extension ˜θn : Qn →ZIrrWn of θn to the coplactic spaceQn (see Theorem 4.14). The coplactic space refer to J¨ollenbeck’s construction revised in [3].
Now, let SP = ⊕n≥0ZWn, Σ′ = ⊕n≥0Σ′(Wn) and Q = ⊕n≥0Qn. Let θ =
⊕n≥0θn and ˜θ=⊕n≥0θ˜n. Aguiar and Mahajan have proved thatSP is naturally a Hopf algebra and that Σ′ is a Hopf subalgebra [1]. We prove here that Q is also a Hopf subalgebra of SP (containing Σ′) and that θ and ˜θ are surjective morphisms of Hopf algebras (see Theorem 5.8). This generalizes similar results in symmetric groups ([17] and [3]), which are parts of combinatorial tools used within the framework of the representation theory of typeA (see for instance [21]).
In the last section of this paper, we give some explicit computations in Σ′(W2) (characters, complete set of orthogonal primitive idempotents, Cartan matrix of Σ′(W2)...).
In the Appendix, P. Baumann and the second author link the above construction with the Specht construction and symmetric functions (see [14]).
Remark. It seems interesting to try to construct a subalgebra Σ′(W) of ZW con- taining Σ(W) and a morphism θ′ : Σ′(W)→ZIrrW for arbitrary Coxeter group W. But it is impossible to do so in a same way as we did for typeC(by extending the generating set). Computations usingCHEVIEprograms show us that it is impos- sible to do so in typeD4and that the reasonable choices inF4fail (we do not obtain a subalgebra!). However, it is possible to do something similar for typeG2. More precisely, let (W, S) be of typeG2. WriteS={s, t}and letS′ ={s, t, sts, tstst}and repeat the procedure described above to obtain a sub-Z-module Σ′(W) ofZW and a morphism θ′ : Σ′(W)→ZIrrW. Then the theorem stated in this introduction also holds in this case. We have rankZΣ′(W) = 8 and rankZKerθ′= 2.
2. Some reflection subgroups of hyperoctahedral groups In this article, we denote [m, n] ={i∈Z|m≤i≤n}={m, m+1, . . . , n−1, n}, for all m ≤n∈ Z, and sign (i) ∈ {±1} the sign of i ∈Z\ {0}. If E is a set, we denote byS(E) the group of permutations on the setE. Ifm∈Z, we often denote bymthe integer−m.
2.1. The hyperoctahedral group. We begin by making clear some notations and definitions concerning thehyperoctahedral groupWn. Denote 1nthe identity of Wn(or 1 if no confusion is possible). We denote byℓt(w) the number of occurrences oftin a reduced decomposition of wand we defineℓs(w) =ℓ(w)−ℓt(w).
It is well-known that Wn acts on the setIn = [1, n]∪[¯n,¯1] by permutations as follows: t= (¯1, 1) and si= (i+ 1, i)(i, i+ 1) for anyi∈[1, n−1]. Through this action, we have
Wn ={w∈S(In)| ∀ i∈In, w(i) =w(i)}.
We often representw∈Wn as the wordw(1)w(2). . . w(n) in examples.
The subgroupWn¯ ={w∈Wn | w([1, n]) = [1, n]} ofWn is naturally identified withSn, the symmetric group of degree n, by restriction of its elements to [1, n].
Note thatWn¯ is generated, as a reflection subgroup ofWn, byS¯n={s1, . . . , sn−1}.
Astandard parabolic subgroup ofWn is a subgroup generated by a subset ofSn
(a parabolic subgroup of Wn is a subgroup conjugate to some standard parabolic subgroup). Note that (Wn¯, S¯n) is a Coxeter group, which is a standard parabolic subgroup ofWn. Ifm≤n, thenSmis naturally identified with a subset ofSn and Wmwill be identified with the standard parabolic subgroup ofWngenerated bySm. Now, we set Tn = {t1, . . . , tn}, with ti as in Introduction. As a permutation of In, note that ti = (i,¯i), then the reflection subgroup Tn generated by Tn is naturally identified with (Z/2Z)n. Therefore Wn = Wn¯ ⋉Tn is just the wreath product of Sn by Z/2Z. If w ∈ Wn, we denote by (wS, wT) the unique pair in Sn×Tn such that w=wSwT. Note thatℓt(w) =ℓt(wT). In this article, we will consider reflection subgroups generated by subsets ofSn′ =Sn∪Tn.
2.2. Root system. Before studying the reflection subgroups generated by subsets of Sn′, let us recall some basic facts about Weyl groups of type C (see [6]). Let us endow Rn with its canonical euclidean structure. Let (e1, . . . , en) denote the canonical basis ofRn: this is an orthonormal basis. Ifα∈Rn\ {0}, we denote by sα the orthogonal reflection such thatsα(α) =−α. Let
Φ+n ={2ei |1≤i≤n} ∪ {ej+νei |ν∈ {1,−1}and 1≤i < j≤n}, Φ−n =−Φ+n and Φn= Φ+n ∪Φ−n. Then Φn is a root system of typeCn and Φ+n is a positive root system of Φn. By sendingttos2e1andsitosei+1−ei(for 1≤i≤n−1), we will identifyWn with the Coxeter group of Φn. Then
∆n ={2e1, e2−e1, e3−e2, . . . , en−en−1}
is the basis of Φn contained in Φ+n and the subsetSn ofWn is naturally identified with the set of simple reflections {sα | α∈ ∆n}. Therefore, for anyw ∈ Wn we have
ℓ(w) =|Φ+n ∩w−1(Φ−n)|;
andℓ(wsα)< ℓ(w) if and only ifw(α)∈Φ−, for allα∈Φ+.
Remark 2.1 - Let w ∈ Wn and let α∈ Φ+n. Then ℓ(wsα)< ℓ(w) if and only if w(α)∈Φ−n. Therefore, ifi∈[1, n−1], then
ℓ(wsi)< ℓ(w)⇔w(i)> w(i+ 1), and, ifj∈[1, n], then
ℓ(wtj)< ℓ(w)⇔w(j)<0.
Therefore, we deduce from the strong exchange condition (see [11,§5.8]) (2.2) ℓt(w) =|{i∈[1, n]|w(i)<0}|.
2.3. Some closed subsystems of Φn. Consider the subsets{s1, t1}and {s1, t2} ofSn′ (n≥2). It is readily seen that these two sets of reflections generate the same reflection subgroup of Wn. This lead us to find a parametrization of subgroups generated by a subset ofSn′.
A signed composition is a sequenceC = (c1, . . . , cr) of non-zero elements ofZ.
The number r is called the length of C. We set |C| = Pr
i=1|ci|. If |C| = n, we say that C is a signed composition of n and we write C||=n. We also define C+= (|c1|, . . . ,|cr|)||=n,C− =−C+andC=−C. We denote by Comp(n) the set
of signed compositions ofn. In particular, any composition is a signed composition (any part is positive). Note that
(2.3) |Comp(n)|= 2.3n−1.
Now, to eachC = (c1, . . . , cr)||=n, we associate a reflection subgroup ofWn as follows: for 1≤i≤r, set
IC(i)=
(IC,+(i) ifci<0, IC,+(i) ∪ −IC,+(i) ifci>0, whereIC,+(i) =
|c1|+· · ·+|ci−1|+ 1,|c1|+· · ·+|ci| . Then WC ={w∈Wn | ∀1≤i≤r, w(IC(i)) =IC(i)} is a reflection subgroup generated by
SC = {sp∈S¯n | |c1|+· · ·+|ci−1|+ 1≤p≤ |c1|+· · ·+|ci| −1}
∪
t|c1|+···+|cj−1|+1∈Tn|cj>0 ⊂Sn′
Therefore,WC≃Wc1× · · · ×Wcr: we denote by (w1, . . . , wr)7→w1× · · · ×wrthe natural isomorphismWc1× · · · ×Wcr
−→∼ WC.
Example. The groupW(2,3,1,3,1) ≃ S2×W3×S1×S3×W1 is generated, as a reflection subgroup ofW10, byS(2,3,1,3,1)={s1}∪{t3, s3, s4}∪{s7, s8}∪{t10} ⊂S10′ . The signed compositionC is said semi-positive if ci ≥ −1 for every i ∈ [1, r].
Note that a composition is a semi-positive composition. We say thatC isnegative ifci<0 for everyi∈[1, r]. We say thatC isparabolic ifci<0 fori∈[2, r]. Note thatC is parabolic if and only ifWC is a standard parabolic subgroup.
Now, let SC′ = Sn′ ∩WC, ΦC = {α ∈ Φn | sα ∈ WC} and Φ+C = ΦC ∩Φ+n. ThenWC is the Weyl group of the closed subsystem ΦC of Φn. Moreover, Φ+C is a positive root system of ΦCand we denote by ∆Cthe basis of ΦCcontained in Φ+C. Note thatSC={sα |α∈∆C}, so (WC, SC) is a Coxeter group.
Let ℓC : WC → Ndenote the length function onWC with respect to SC. Let wC denote the longest element ofWC with respect to ℓC. If C is a composition, we denote byσC the longest element ofSC=WCwith respect toℓC(which is the restriction of ℓ to SC). In other words, σC = wC. In particular, wn (resp. σn) denotes the longest element ofWn (resp. Sn).
WriteTC=Tn∩WC andTC=Tn∩WC, then observe that (2.4) WC =WC−⋉TC=SC+⋉TC.
Remarks. (1) This class of reflection subgroups contains the standard parabolic subgroups, since Sn ⊂ Sn′. But it contains also some other subgroups which are not parabolic (consider the subgroup generated by {t1, t2} as example). In other words, it may happen that ∆C 6⊂∆n. In fact, ∆C ⊂ ∆n if and only if WC is a standard parabolic subgroup ofWn.
(2) If WC is not a standard parabolic subgroup of Wn, then ℓC is not the re- striction ofℓto WC.
We close this subsection by an easy characterization of the subsetsSC′ : Lemma 2.5. Let X be a subset ofSn′. Then the following are equivalent:
(1) < X >∩Sn′ =X.
(2) X∩Tn is stable under conjugation by< X >.
(3) X∩Tn is stable under conjugation by< X∩Sn¯ >.
(4) There exists a signed composition C of nsuch that X=SC′ .
Corollary 2.6. Let w ∈ Wn and let C||=n. If wSC′ ⊂ Sn′, then there exists a (unique) signed compositionD such that wSC′ =SD′ .
Proof. Indeed,wSC′ ∩Tn=w(S′C∩Tn) and wSC′ ∩S¯n=w(SC′ ∩Sn¯).
2.4. Orbits of closed subsystems ofΦn. In this subsection, we determine when two subgroups WC and WD of Wn are conjugated. A bipartition of n is a pair λ= (λ+, λ−) of partitions such that|λ|:=|λ+|+|λ−|=n. We writeλnto say that λis a bipartition ofn, and the set of bipartitions ofn is denoted by Bip(n).
It is well-known that the conjugacy classes ofWn are in bijection with Bip(n) (see [9, 14]). We define ˆλas the signed composition ofnobtained by concatenation of λ+ and−λ−. The map Bip(n)→Comp(n),λ7→λˆ is injective.
Now, let C be a signed composition of n. We defineλ(C) = (λ+, λ−) as the bipartition of n such that λ+ (resp. λ−) is obtained from C by reordering in decreasing order the positive parts of C (resp. the absolute value of the negative parts ofC). One can easily check that the map
λ: Comp(n)−→Bip(n)
is surjective (indeed, ifλ∈Bip(n), thenλ(ˆλ) =λ) and that the following proposi- tion holds:
Proposition 2.7. Let C, D||=n, then WC and WD are conjugate in Wn if and only ifλ(C) =λ(D). If Ψis a closed subsystem of Φn, then there exists a unique bipartition λofn and somew∈Wn such thatΨ =w(Φˆλ).
LetC, D||=n, then we write C⊂D ifWC ⊂WD. Moreover,C,C′ ⊂D and if WC andWC′ are conjugate underWD, then we write C≡DC′.
2.5. Distinguished coset representatives. LetC||=n, then XC={x∈Wn | ∀w∈WC, ℓ(xw)≥ℓ(x)}
is a distinguished set ofminimal coset representativesforWn/WC (see proposition below). It is readily seen that
XC = {w∈Wn |w(Φ+C)⊂Φ+n}
= {w∈Wn | ∀ α∈∆C, w(α)∈Φ+n}.
Finally
XC ={w∈Wn | ∀r∈SC, ℓ(wr)> ℓ(w)}.
We need a relative notion: ifD||=n such thatC ⊂D, the setXCD=XC∩WD
is a distinguished set of minimal coset representatives forWD/WC. IfD= (n) we writeXCn instead ofXC(n).
Proposition 2.8. Let C||=n, then:
(a) The mapXC×WC→Wn,(x, w)7→xw is bijective.
(b) If C⊂D, then the map XD×XCD→XC, (x, y)7→xy is bijective.
(c) If x ∈ XC and w ∈ WC, then ℓt(xwx−1) ≥ ℓt(w). Consequently, Sn∩
xWC=Sn∩xSC+.
Proof. (a) is stated, in a general case, in [7, Proposition 3.1]. (b) follows easily from (a). Let us now prove (c). Letx∈XC andw∈WC. LetI ={i∈In |w(i)<0}
and J ={i ∈In | xwx−1(i)<0}, then ℓt(w) =|I| and ℓt(xwx−1) =|J|, by 2.2.
Now leti∈I, thenti∈WC, soℓt(xti)> ℓt(x). In other words,x(i)>0. Now, we havexwx−1(x(i)) =xw(i). But, w(i)<0 andt−w(i)=wtiw−1∈WC. Therefore, x(−w(i)) =−xw(i)>0. This shows thatx(i)∈J. So, the mapI→J, i7→x(i) is well-defined and clearly injective, implying|I| ≤ |J|as desired.
The last assertion of this proposition follows easily from this inequality and from
the fact thatSC+ ={w∈WC | ℓt(w) = 0}.
Proposition 2.9. Let C||=n andx ∈XC be such that xSC′ ⊂Sn′. Let D be the unique signed composition of n such that xSC′ = SD′ (see Corollary 2.6). Then XC =XDx.
Proof. By symmetry, it is sufficient to prove that, ifw∈XD, then wx∈XC. Let α∈Φ+C. Then, since x∈XC, we havex(α)∈Φ+n ∩xΦC = Φ+D. Sow(x(α))∈Φ+n
sincew∈XD. Sowx∈XC.
2.6. Maximal element in XC. It turns out that, for every signed composition C of n, XC contains a unique element of maximal length (see Proposition 2.12).
First, note the following two examples:
(1) ifC is parabolic, it is well-known thatℓC is the restriction ofℓand that, for all (x, w)∈XC×WC, we have
ℓ(xw) =ℓ(x) +ℓ(w) In particular,wnwC is the longest element ofXC(see [9]);
(2) letC be a composition ofn, thenWC is not in general a standard parabolic subgroup of Wn. However, since WC contains Tn, XC is contained in Sn. This shows that
XC=XC¯n =XC∩Sn.
In particular,XC contains a unique element of maximal length: this isσnσC; Now, letk and l be two non-zero natural numbers such thatk+l =n. Then Wk,l is not a parabolic subgroup of Wn. However, Wk,¯l is a standard parabolic subgroup of Wn andWk,¯l ⊂Wk,l. So Xk,l ⊂Xk,¯l. So, if x∈Xk,l and w∈Wk,¯l, then
(2.10) ℓ(xw) =ℓ(x) +ℓ(w).
This applies for instance ifw∈Wk ⊂Wk,¯l.
Then, we need to introduce a decomposition of XC using Proposition 2.8 (b).
WriteC = (c1, . . . , cr)||=n. We set
XC,i=X(|c(|c11|+···+|c|+···+|ci|,ci+1,...,cr)
i−1|,ci,...,cr). Then the map
XC,r× · · · ×XC,2×XC,1 −→ XC
(xr, . . . , x2, x1) 7−→ xr. . . x2x1
is bijective by Proposition 2.8 (b). Moreover, by 2.10, we have (2.11) ℓ(xr. . . x2x1) =ℓ(xr) +· · ·+ℓ(x2) +ℓ(x1)
for every (xr, . . . , x2, x1) ∈ XC,r× · · · ×XC,2×XC,1. For every i ∈ [1, r], XC,i
contains a unique element of maximal length (see (1)-(2) above). Let us denote it byηC,i. We set:
ηC=ηC,r. . . ηC,2ηC,1. Then, by 2.11, we have
Proposition 2.12. Let C||=n, then ηC is the unique element of XC of maximal length.
2.7. Double cosets representatives. IfCandDare two signed compositions of n, we set
XCD =XC−1∩XD.
Proposition 2.13. LetC andD be two signed composition ofnand letd∈XCD. Then:
(a) There exists a unique signed compositionEof nsuch thatSE′ =S′C∩dSD′ . It will be denoted byC∩dD or dD∩C. We have (C∩dD)−=C−∩dD−. (b) WC∩dWD=WC∩dD and WC∩dSD′ =SC′ ∩dWD=SC∩′ dD.
(c) If w∈WC∩dD, thenℓt(w) =ℓt(d−1wd).
(d) Ifw∈WCdWD, then there exists a unique pair (x, y)∈XC∩C dD×WDsuch that w=xdy.
(e) Let(x, y)∈XC∩C dD×WD, thenℓ(xdy)≥ℓ(xS)+ℓt(x)+ℓ(d)+ℓ(yS)+ℓt(y).
(f) dis the unique element ofWCdWD of minimal length.
Proof. (a) follows immediately from Lemma 2.5 (equivalence between (3) and (4)).
(b) It is clear that WE ⊂WC∩dWD. Let us show the reverse inclusion. Let w∈WC∩dWD. We will show by induction onℓt(w) thatw∈WE. Ifℓt(w) = 0, then we see from Proposition 2.8 (d) that w∈SC+∩dSD+ =SE+ by definition ofE+.
Assume now thatℓt(w)>0 and that, ifw′ ∈WC∩dWD is such thatℓt(w′)<
ℓt(w), then w′ ∈ WE. Sinceℓt(w)>0, there existsi ∈[1, n] such thatw(i)<0.
In particular, ti ∈TC. By the same argument as in the proof of Proposition 2.8 (d), we have that ti ∈ dWD. So, ti ∈ TC∩dTD =TE. Now, letw′ =wti. Then ti∈WE,w′∈WC∩dWD andℓt(w′) =ℓt(w)−1. So, by the induction hypothesis, w′ ∈WE, sow∈WE.
The other assertions of (b) follow easily.
(c) Let w = σ1. . . σl be a reduced decomposition of w with respect to SC. Then d−1wd = (d−1σ1d). . .(d−1σld). But d−1σid ∈ d−1(S′C∩dSD′ ) =S′d−1
C∩D, so ℓt(d−1σid) = ℓt(σi). Since ℓt(w) = ℓt(σ1) +· · ·+ℓt(σl), we see that ℓt(w) ≥ ℓt(d−1wd). By symmetry, we obtain the reverse inequality.
(d) Letw∈WCdWD. Let us writew=adb, witha∈WC andb∈WD. We then writea=xa′ withx∈XC∩C dD anda′ ∈SC∩dD. Then d−1a′d∈Wd−1C∩D⊂WD. Writey = (d−1a′d)b. Then (x, y)∈XC∩C dD×WD andw=xdy.
Now let (x′, y′)∈XC∩C dD×WDsuch thatw=x′dy′. Thenx′−1x=d(yy′−1)d−1. Sox′−1x∈WC∩dD, that isxWC =x′WC. Sox=x′ andy=y′.
(e) Let (x, y)∈XC∩C dD×WD. We will show by induction onℓt(x) +ℓt(y) that ℓ(xdy)≥ℓ(xS) +ℓt(x) +ℓ(d) +ℓ(yS) +ℓt(y).
If ℓt(x) =ℓt(y) = 0, then x∈XCC−−∩d(D−), y ∈SD+ and d∈XC−,D−. So, by [2, Lemma 2], we haveℓ(xdy) =ℓ(xS) +ℓ(d) +ℓ(yS), as desired.
Now, let us assume that ℓt(x) +ℓt(y) > 0 and that the result holds for every pair (x′, y′)∈XC∩C dD×WD such thatℓt(x′) +ℓt(y′)< ℓt(x) +ℓt(y). By symmetry, and using (c), we can assume that ℓt(y) > 0. So there exists i ∈ In such that y(i)<0. Lety′=yti. Then ti ∈TD,ℓ(yS) =ℓ(y′S),ℓt(y′) =ℓt(y)−1. Therefore, by induction hypothesis, we have
ℓ(xdy′)≥ℓ(xS) +ℓt(x) +ℓ(d) +ℓ(yS) +ℓt(y)−1.
It is now enough to show thatℓ(xdy′ti)> ℓ(xdy′), that isxdy′(i)>0. Note that y′(i)>0 and thatty′(i)=y′tiy′−1∈WD. So the result follows from the following lemma:
Lemma 2.14. If d∈XCD, if x∈XC∩D dD and ifj∈[1, n] is such thattj∈TD, thenxd(j)>0.
Proof. Sincetj ∈WD and d∈XD, we haved(j)>0. Two cases may occur. Iftd(j)∈TC, then td(j)=dtjd−1∈TC∩dD. Therefore, x(d(j))>0 sincex∈XC∩C dD. Iftd(j)6∈TC, thenx(d(j))>0 since
x∈WC=SC+⋉TC.
(f) follows immediately from (e).
Remark 2.15 - Let C and D be two signed compositions ofn and let d∈XCD. Thend−1∈XDC and, by Proposition 2.9, we have that
XC∩dDd=Xd−1C∩D.
Corollary 2.16. The map XCD →WC\Wn/WD is bijective.
Proof. The proposition 2.13 (f) shows that the map is injective. The surjectivity follows from the fact that, ifw∈Wn is an element of minimal length in WCwWD,
thenw∈XCD.
Corollary 2.17. If C is parabolic or if D is semi-positive, then XD= a
d∈XCD
XC∩C dDd.
Proof. It follows from Corollary 2.16 that
|XD|=|Wn/WD|= X
d∈XCD
|WCdWD/WD|= X
d∈XCD
|XC∩dD|,
the last equality following from Proposition 2.13 (d). So, it remains to show that, ifd∈XCD and ifx∈XC∩C dD, thenxd∈XD.
Assume that we have founds∈SDsuch thatℓ(xds)< ℓ(xd). Ifs∈TD, thens= ti for somi∈In. But, by Lemma 2.14,xd(i)>0, soℓ(xdti)> ℓ(xd), contradicting our hypothesis. Therefore, s ∈ SD−, that is s = si for some i ∈ [1, n−1]. If C is parabolic, C∩dD is also parabolic. Therefore, ℓ(xds) > ℓ(xd) which is a contradiction, soD is semi-positive. Therefore, we have thatti andti+1 belong to
TD. Thus, by Lemma 2.14, we havexd(i)>0 andxd(i+ 1)>0. Moreover, since ℓ(xdsi)< ℓ(xd), we have
(∗) 0< xd(i+ 1)< xd(i).
But, sinced∈XD, we haved(i+ 1)> d(i). So, by Proposition 2.13 (b), we have that dsid−1 ∈ SC∩dD. Thus ℓ(x(dsid−1)) > ℓ(x) because x ∈ XC∩C dD. In other
words,xd(i+ 1)> xd(i). This contradicts (∗).
IfE is a signed composition ofn such that C⊂E andD ⊂E, we setXCDE = XCD∩WE.
Example. It is not true in general that XD = `
d∈XCDXC∩C dDd. This is false, if n=k+l withk,l≥1,C= (k, l) andD= (n). See Example 2.25 for precisions.
In [2], the authors has given a proof of the Solomon theorem using tools which sound like the above results. Here, we cannot translate their proof because of the complexity of the decomposition ofXD(which involve negative coefficients).
2.8. A partition of Wn. IfC= (c1, . . . , cr) is a signed composition ofn, we set AC={s|c1|+···+|ci| |i∈[1, r] andci<0 andci+1>0}
and AC=SC′ a
AC.
As example,A(1,¯3,¯1,2,¯1,1)={s5, s8}. Note thatAC =AD if and only ifC =D. If w∈Wn, then we definethe ascent set ofw:
Un′(w) ={s∈Sn′ |ℓ(ws)> ℓ(w)}.
Finally, following Mantaci-Reutenauer, we associate to each element w ∈ Wn a signed compositionC(w) as follows. First, letC+(w) denote the biggest composi- tion (for the order⊂) ofnsuch that, for every 1≤i≤r, the mapw:IC(i)+(w)→In
is increasing and has constant sign. Now, we defineνi= sign (w(j)) forj∈IC(i)+(w). Thedescent compositionofwisC(w) = (ν1c+1, . . . , νrc+r).
Example. C(9.|¯3¯2¯1.{z¯4.58.¯6.7}
∈W9
) = (1,¯3,¯1,2,¯1,1)||=9.
The following proposition is easy to check (see Remark 2.1):
Proposition 2.18. Ifw∈Wn, thenUn′(w) =AC(w).
Remark. Mantaci and Reutenauer have defined the descent shapeof a signed per- mutation [16]. It is a signed composition defined similarly than descent composition except that the absolute value of the letters inui must be in increasing order. For instance, the descent shape of 9.¯3.¯2.¯1¯4.58.¯6.7 is (1,¯1,¯1,¯2,2,¯1,1).
Example 2.19 - Let n′ be a non-zero natural number,n′ < nand let c ∈Zsuch that n−n′ =|c|. Letw ∈Wn′ ⊂Wn and writeC(w) = (c1, . . . , cr)||=n′. Then C(η(n′,c)w) = (c1, . . . , cr, c). Consequently, ifC||=n, an easy induction argument shows thatC(ηC) =C.
We have then defined a surjective map
C:Wn−→Comp(n)
whose fibers are equal to those of the applicationUn′ :Wn → P(Sn′). The surjec- tivity follows from Example 2.19. IfC||=n, we define
YC={w∈Wn |C(w) =C}.
Then
Wn= a
C||=n
YC.
Example 2.20 - We haveYn ={1n},Yn¯={σnwn},Y(1,...,1)={σn}andY(¯1,...,¯1)= {wn}.
First, note the following elementary facts.
Lemma 2.21. Let C andD be two signed compositions ofn. Then:
(a) If YC∩XD6=∅, thenYC⊂XD. (b) ηC ∈YC andYC ⊂XC.
Proof. (a) Ifw ∈Wn, then w ∈XD if and only ifUn′(w) contains SD′ . Since the mapw7→ Un′(w) is constant onYC(see Proposition 2.18), (a) follows.
(b) By Example 2.19, we haveηC∈YC∩XC. Therefore, by (a),YC ⊂XC. We then define a relation← between signed composition of nas follow. If C, D||=n, we writeC←DifYD⊂XC. We denote by 4 the transitive closure of the relation←. It follows from Lemma 2.21 (a) that
(2.22) XC= a
C←D
YD.
Example 2.23 - Letw∈Wn. By Remark 2.1,w∈Xn¯ if and only if the sequence (w(1), w(2), . . . , w(n)) of elements ofIn is strictly increasing (see Remark 2.1). So there exists a unique k ∈ {0,1,2, . . . , n} such that w(i) >0 if and only if i > k.
Note thatk=ℓt(w). Leti1<· · ·< ik be the sequence of elements ofIn such that (w(1), . . . , w(k)) = (¯ik, . . . ,¯i1). Thenw=ri1ri2. . . rik where, if 1≤i≤n, we set ri=si−1. . . s2s1t. Note thatC(w) = (¯k, n−k). Therefore,
X¯n={ri1ri2. . . rik |0≤k≤nand 1≤i1< i2<· · ·< ik≤n}.
Note thatℓ(ri1ri2. . . rik) =i1+i2+· · ·+ik andℓt(ri1ri2. . . rik) =k. We get Xn¯ = a
0≤k≤n
Y(¯k,n−k), and, for everyk∈ {0,1,2, . . . , n}, we have
Y(¯k,n−k)={ri1ri2. . . rik |1≤i1< i2<· · ·< ik≤n}.
This shows that (¯n)←(¯k, n−k).
Proposition 2.24. Let C andD be two signed compositions ofn. Then:
(a) C←C.
(b) If C⊂D, thenC←D.
(c) 4 is an order onComp(n).
Proof. (a) follows immediately from Lemma 2.21 (b).
(b) IfC⊂D, thenXD⊂XC. But, by Lemma 2.21 (b), we haveYD⊂XD. So C←D.
(c) LetaC =ℓ(µC). By (a), 4 is reflexive. By definition, it is transitive. So it is sufficient to show that it is antisymmetric. But it follows from Lemma 2.21 (b) that:
•IfC←D, thenaD≤aC.
•IfC←D and ifaC=aD, thenC=D.
The assertion (c) now follows easily from these two remarks.
Example 2.25 - IfC= (c1, . . . , cr) is a composition ofn(not a signed composition), we will prove that
Xn¯= a
0≤m2≤c2
0≤m3≤c3
. . .
0≤mr≤cr
X(¯Cc1,m2,c2−m2,...,mr,cr−mr)Y(¯(¯cc11,,mm¯22,c,c22−m−m22,...,,...,mm¯rr,c,crr−m−mrr))σ−1C,m2,...,mr,
whereσC,m2,...,mr ∈Sn satisfies
σC,m2,...,mr(S(c′ 1,m2,c2−m2,...,mr,cr−mr))⊂S′n and σC,m2,...,mr∈X(c1,m2,c2−m2,...,mr,cr−mr).
By an easy induction argument, it is sufficient to prove it wheneverr= 2. In other words, we want to prove that, ifk+l=nwith k,l≥0, then
(∗) Xn¯ = a
0≤m≤l
X(¯(¯k,m,l−m)k,¯l) Y(¯(¯k,k,m,l−m)m,l−m)¯ σk,l,m−1 ,
whereσk,l,m∈Sn satisfiesσk,l,m(Sk,m,l−m′ )⊂Sn′ andσk,l,m∈X(k,m,l−m). But, if 0≤m≤l, we set
σk,l,m(i) =
m+i if 1≤i≤k,
i−k ifk+ 1≤i≤k+m, i ifk+m+ 1≤i≤n,
and one can easily check that (∗) holds. Moreover, sinceSk,m,l−m′ =Sn′\{sk, sk+m}, we get thatσk,l,m(Sk,m,l−m′ )⊂S′n andσk,l,m∈X(k,m,l−m).
3. Generalized descent algebra
3.1. Definition. If C and D are two signed compositions ofn such thatC⊂D, we set
xDC = X
w∈XCD
w ∈ZWD
and yDC = X
w∈YCD
w ∈ZWD. Now, let
Σ′(WD) = ⊕
C⊂D
ZyDC ⊂ZWD.
Note that
Σ′(WD) = ⊕
C⊂DZxDC by 2.22 and Proposition 2.24. We define
θD: Σ′(WD)−→ZIrrWD
as the uniqueZ-linear map such that
θD(xDC) = IndWWDC 1C
for every C ⊂D. Here, 1C is the trivial character ofWC. We denote by εD the sign character ofWD.
Notation. If D = (n), we set xDC = xC, yCD = yC for simplification. If E is Z- module, we denote by QE the Q-vector space Q⊗ZE. We denote by θD,Q the extension ofθD toQΣ′(WD) byQ-linearity.
Remark. Σ′(Wn) contains the Solomon descent algebras of Wn and Sn. More- over, Σ′(Wn) is precisely the Mantaci-Reutenauer algebra which is, by definition, generated byyD=yD(n), for allD||=n.
3.2. First properties of θD. By the Mackey formula for product of induced characters and by Proposition 2.13, we have that
(3.1) θn(xC)θn(xD) = X
d∈XCD
θn(xd−1
C∩D).
Example 3.2 - IfC is parabolic orD is semi-positive, then, by Corollary 2.17, we have
xD= X
d∈XCD
xCC∩dDd.
Therefore, by Proposition 2.8 (b) and Remark 2.15, we get xCxD= X
d∈XCD
xd−1
C∩D.
SoxCxD∈Σ′(Wn) and, by 3.1,θn(xCxD) =θn(xC)θn(xD).
Before starting the proof of the fact that Σ′(WD) is a subalgebra ofZWD and thatθDis a morphism of algebras, we need the following result, which will be useful for arguing by induction. IfC ⊂D, the transitivity of induction and Proposition 2.8 (b) show that the diagram
(3.3)
Σ′(WC) xDC.
//
θC
Σ′(WD)
θD
ZIrrWC
IndWWDC
//ZIrrWD
is commutative.
Now, let pD : WD → SD+ be the canonical projection. It induces an injec- tive morphism of Z-algebrasp∗D : ZIrrSD+ → ZIrrWD. Moreover, the algebra
Σ′(SD+) coincides with the usual descent algebra in symmetric groups and is con- tained in Σ′(WD). Also, the diagram
(3.4)
Σ′(SD+) //
θD−
Σ′(WD)
θD
ZIrrSD+ p∗D
//ZIrrWD
is commutative.
Example 3.5 - We havey(¯1,...,¯1)=wn,yn¯ =wnσn=σnwn, yn = 1 andy(1,...,1)= σn. It is well-known [18] that y(¯1,...,¯1) belongs to the classical descent algebra of Wn and that
(a) θn(wn) =εn.
On the other hand,
(b) θn(1n) = 1(n).
Also, by the commutativity of the diagram 3.4 and as above, we have
(c) θn(σn) =γn,
whereγn =p∗nεn¯. Finally,wn is aZ-linear combination of xC, whereC runs over the parabolic compositions of n. Therefore, by Example 3.2, we have, for every x∈Σ′(Wn),
(d) θn(wnx) =θn(wn)θn(x) =εnθn(x).
In particular,
(e) θn(y¯n) =θn(wnσn) =εnγn.
So we have obtained the four linear characters of Wn as images by θn of explicit elements of Σ′(Wn).
Let degD :ZIrrWD →Zbe theZ-linear map sending an irreducible character ofWD to its degree. It is a morphism ofZ-algebras. Let augD:ZWD →Zbe the augmentation morphism, then it is clear that the diagram
(3.6)
Σ′(WD) θD
//
augD
$$
HH HH H HH HH HH HH HH H HH HH
ZIrrWD
degD
Z
is commutative.
3.3. Main result. We are now ready to prove that Σ′(WD) is a Z-subalgebra of ZWD and thatθD is a surjective morphism of algebras.
Theorem 3.7. Let D be a signed composition ofn, then:
(a) Σ′(WD)is aZ-subalgebra of ZWD; (b) θD is a morphism of algebra;
(c) θD is surjective and KerθD=L
C,C′⊂D C≡DC′
Z(xDC−xDC′);
(d) KerθD,Q is the radical of the algebraQΣ′(WD). Moreover, QΣ′(WD) is a split algebra whose largest semisimple quotient is commutative. In particu- lar, all its simple modules are of dimension1.
Proof. We want to prove the theorem by induction on |WD|. By taking direct products, we may therefore assume that D= (n) orD= (¯n). IfD= (¯n), then it is well-known that (a), (b), (c) and (d) hold. So we may assume thatD= (n) and that (a), (b), (c) and (d) hold for every signed compositionD′ ofndifferent from (n).
(a) and (b): LetA and B be two signed compositions of n. We want to prove that xAxB ∈Σ′(Wn) and thatθn(xAxB) =θn(xA)θn(xB). IfA is parabolic or B is semi-positive, then this is just Example 3.2. So we may assume that A is not parabolic andB is not semi-positive.
First, note that B ⊂B+ and that B+ is semi-positive. Therefore, by Proposi- tion 2.8(b) and Example 3.2, we have
xAxB =xAxB+xBB+ = X
d∈XA,B+
xd−1A∩B+xBB+=xB+ X
d∈XA,B+
xBd−1+A∩B+xBB+. Assume first thatB+6= (n). Then, by induction hypothesis,
X
d∈XA,B+
xBd−1+A∩B+xBB+ ∈Σ′(SB+) and
θB+ X
d∈XA,B+
xBd−+1
A∩B+xBB+
= X
d∈XA,B+
θB+(xBd−+1
A∩B+)θB+(xBB+).
Therefore, by 3.3 and 3.4, xAxB ∈xB+Σ′(SB+)⊂Σ′(Sn)⊂Σ′(Wn) and, by 3.3 and by the Mackey formula for tensor product, we get
θn(xAxB) = IndWWn
B+
X
d∈XA,B+
θB+(xBd−1+A∩B+)θB+(xBB+)
= θn(xA) IndWWn
B+θB+(xBB+)
= θn(xA)θn(xB), as desired.
Therefore, it remains to consider the case whereB+ = (n). In particular,B = (n) or (¯n). SinceB is not semi-positive, we haveB = (¯n). By Example 2.25, we have
xn¯=xAA+−+ X
D⊂A+
aDxAD+(σD−1−1)
