Algebra
A Note on the Grothendieck ring of the symmetric group
Cédric Bonnafé
Laboratoire de mathématiques de Besançon, (CNRS – UMR 6623), université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France
Received 6 December 2005; accepted 21 February 2006 Available online 20 March 2006
Presented by Michèle Vergne
Abstract
Letpbe a prime number and letnbe a non-zero natural number. We compute the descending Loewy series of the algebra Rn/pRn, whereRndenotes the ring of virtual ordinary characters of the symmetric groupSn.To cite this article: C. Bonnafé, C. R. Acad. Sci. Paris, Ser. I 342 (2006).
2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.
Résumé
Une note sur l’anneau de Grothendieck du groupe symétrique.Soitpun nombre premier et soitnun entier naturel non nul.
Nous calculons la série de Loewy descendante de l’algèbreRn/pRn, oùRndésigne l’anneau des caractères virtuels ordinaires du groupe symétriqueSn.Pour citer cet article : C. Bonnafé, C. R. Acad. Sci. Paris, Ser. I 342 (2006).
2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.
Version française abrégée
Soitpun nombre premier et soitnun entier naturel non nul. SoitRnl’anneau des caractères virtuels ordinaires du groupe symétriqueSn et posonsRn=Rn/pRn. AlorsRnest uneFp-algèbre commutative de dimension finie égale à|Part(n)|, où Part(n)désigne l’ensemble des partitions den. Le but de cette Note est de décrire la série de Loewy descendante de cette algèbre, c’est-à-dire la suite des puissances de son radical : nous déterminons une base de chacune de ces puissances.
Pour énoncer le résultat, nous aurons besoin de quelques notations. Tout d’abord, siϕ∈Rn, on noteϕ¯ son image dansRn. Siλ=(λ1, λ2, . . . , λr)∈Part(n), on noteSλle sous-groupe de Young deSnengendré par les transpositions (i, i+1)satisfaisanti=k
j=1λj pour toutk∈ {1,2, . . . , r}. Ce groupeSλ est canoniquement isomorphe àSλ1× Sλ2× · · · ×Sλr. On poseϕλ=IndSSn
λ1λ, où 1λdésigne le caractère trivial deSλ. Sii1, on noteri(λ)le nombre d’apparitions deicomme part deλ. On pose :
πp(λ)=
i1
ri(λ) p
,
E-mail address:bonnafe@math.univ-fcomte.fr (C. Bonnafé).
1631-073X/$ – see front matter 2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.
doi:10.1016/j.crma.2006.02.028
où[x]désigne la partie entière dex. Posons Part(i)p (n)=
λ∈Part(n)|πp(λ)i .
Un vieux résultat de Frobenius nous dit que(ϕλ)λ∈Part(n) est uneZ-base deRn. La série de Loewy descendante de Rnadmet une decription très simple dans cette base :
Théorème A.Sii0, alors(ϕ¯λ)λ∈Part(i)
p(n)est uneFp-base de(RadRn)i.
Rappelons que la longueur de Loewy d’un anneauRest le plus petiti∈N∗∪ {∞}tel que(RadR)i=0.
Corollaire B.La longueur de Loewy deRnest[n/p] +1.
1. Introduction
Letpbe a prime number and letnbe a non-zero natural number. LetFpbe the finite field withpelements and let Snbe the symmetric group of degreen. LetRndenote the ring of virtual ordinary characters of the symmetric group Sn and letRn=Fp⊗ZRn=Rn/pRn. The aim of this paper is to determine the descending Loewy series of the Fp-algebraRn(see Theorem A). In particular, we deduce that the Loewy length ofRnis[n/p] +1 (see Corollary B).
Here, ifx is a real number,[x]denotes the uniquer∈Zsuch thatrx < r+1.
Let us introduce some notation. If ϕ∈Rn, we denote byϕ¯ its image inRn. The radical ofRn is denoted by RadRn. IfXandY are two subspaces ofRn, we denote byXY the subspace ofRngenerated by the elements of the formxy, withx∈Xandy∈Y.
2. Statement
2.1. Compositions, partitions
Acompositionis a finite sequenceλ=(λ1, . . . , λr)of non-zero natural numbers. We set|λ| =λ1+· · ·+λrand we say thatλis acomposition of|λ|. Theλi’s are called thepartsofλ. If moreoverλ1λ2· · ·λr, we say thatλis apartition of |λ|. The set of compositions (resp. partitions) ofnis denoted by Comp(n)(resp. Part(n)). We denote by λˆ the partition ofnobtained fromλby reordering its parts. So Part(n)⊂Comp(n)and Comp(n)→Part(n),λ→ ˆλ is surjective. If 1in, we denote byri(λ)the number of occurrences ofias a part ofλ. We set
πp(λ)= n
i=1
ri(λ) p
.
Recall thatλis calledp-regular(resp.p-singular) if and only ifπp(λ)=0 (resp.πp(λ)1). Note also thatπp(λ)∈ {0,1,2, . . . ,[n/p]}and thatπp(ˆλ)=πp(λ). Finally, ifi0, we set
Part(p)i (n)=
λ∈Part(n)|πp(λ)i .
2.2. Young subgroups
For 1in−1, letsi=(i, i+1)∈Sn. LetSn= {s1, s2, . . . , sn−1}. Then(Sn, Sn)is a Coxeter group (see for instance [2, Proposition 1.4.7]). We denote by:Sn→Nthe associated length function. Ifλ=(λ1, . . . , λr)∈ Comp(n), we set
Sλ= {si| ∀1jr, i=λ1+ · · · +λj}.
LetSλ= Sλ. Then(Sλ, Sλ)is a Coxeter group (see for instance [2, 1.2.9]): it is a standard parabolic subgroup of Snwhich is canonically isomorphic toSλ1× · · · ×Sλr. Note that
SλandSµare conjugate inSnif and only if λˆ= ˆµ. (1)
We writeλ⊂µifSλ⊂Sµand we writeλµifSλis contained in a subgroup ofSnconjugate toSµ. Then⊂is an order on Comp(n)andis a preorder on Comp(n)which becomes an order when restricted to Part(n).
LetXλ= {w∈Sn| ∀x∈Sλ, (wx)(w)}. ThenXλis a cross-section ofSn/Sλ(see for instance [2, Proposi- tion 2.1.1]). Now, letNλ=NSn(Sλ)andW (λ)=Nλ∩Xλ. ThenW (λ)is a subgroup ofNλandNλ=W (λ)Sλ (see for instance [2, Proposition 2.1.15]). Note that
W (λ)Sr1(λ)× · · · ×Srn(λ). (2)
Recall that, for a finite groupG, thep-rankofGis the maximal rank of an elementary Abelianp-subgroup ofG. For instance,[n/p]is thep-rank ofSn. So
πp(λ)is thep-rank ofW (λ). (3)
Ifλ,µ∈Comp(n), we set Xλµ=(Xλ)−1∩Xµ.
ThenXλµis a cross-section ofSλ\Sn/Sµ(see for instance [2, Proposition 2.1.7]). Moreover, ifd∈Xλµ, there exists a unique compositionνofnsuch thatSλ∩dSµ=Sν (see for instance [2, Theorem 2.1.12]). This composition will be denoted byλ∩dµor bydµ∩λ.
2.3. The ringRn
If λ∈Comp(n), we denote by 1λ the trivial character of Sλ and we setϕλ=IndSSn
λ1λ. Then, by (1), we have ϕλ=ϕλˆ. We recall the following well-known old result of Frobenius (see for instance [2, Theorem 5.4.5(b)]):
(ϕλ)λ∈Part(n)is aZ-basis of Rn. (4)
Moreover, by the Mackey formula for tensor product of induced characters (see for instance [1, Theorem 10.18]), we have
ϕλϕµ=
d∈Xλµ
ϕλ∩dµ=
d∈Xλµ
ϕ
λ∩dµ. (5)
Let us give another form of (5). Ifd∈Xλµ, we define∆d:Nλ∩dNµ→Nλ×Nµ,w→(w, d−1wd). Let∆¯d:Nλ∩
dNµ→W (λ)×W (µ)be the composition of∆dwith the canonical projectionNλ×Nµ→W (λ)×W (µ). Then the kernel of∆¯disSλ∩dµ, so∆¯dinduces an injective morphism∆˜d:W (λ, µ, d) →W (λ)×W (µ), whereW (λ, µ, d)= (Nλ∩dNµ)/Sλ∩dµ. Now,W (λ)×W (µ) acts on Sλ\Sn/Sµ and the stabilizer of SλdSµ inW (λ)×W (µ) is
∆˜d(W (λ, µ, d)). Moreover, if d is an element ofXλµ such that SλdSµ andSλdSµ are in the same (W (λ)× W (µ))-orbit, thenSλ∩dµandSλ∩dµare conjugate inNλ. Therefore,
ϕλϕµ=
d∈Xλµ
|W (λ)| · |W (µ)|
|W (λ, µ, d)| ϕλ∩dµ, (6)
whereXλµ denotes a cross-section ofNλ\Sn/Nµcontained inXλµ. 2.4. The descending Loewy series ofRn
We can now state the main results of this Note.
Theorem A.Ifi0, we have(RadRn)i=
λ∈Part(p)i (n)Fpϕ¯λ. Corollary B.The Loewy length ofRnis[n/p] +1.
Corollary B follows immediately from Theorem A. The end of this paper is devoted to the proof of Theorem A.
3. Proof of Theorem A LetR(i)n =
λ∈Part(p)i (n)Fpϕ¯λ. Note that
0= R(n[n/p]+1)⊂ R(n[n/p])⊂ · · · ⊂ R(1)n ⊂ R(0)n = Rn. Let us first prove the following fact:
(♣) If i, j0,thenR(i)n R(j )n ⊂ R(in+j ).
Proof of (♣). Let λ andµ be two compositions ofn such thatπp(λ)i and πp(µ)j. Let d∈Xλµ be such that p does not divide |W (λ)|W (λ,µ,d)|·|W (µ)||. By (6), we only need to prove that this implies thatπp(λ∩dµ)i+j. But our assumption ond means that∆˜d(W (λ, µ, d))contains a Sylowp-subgroup ofW (λ)×W (µ). In particular, the p-rank ofW (λ, µ, d)is greater than or equal to thep-rank ofW (λ)×W (µ). By (3), this means that thep-rank of W (λ, µ, d)isi+j. SinceW (λ, µ, d)is a subgroup ofW (λ∩dµ), we get that thep-rank ofW (λ∩dµ)isi+j. In other words, again by (3), we haveπp(λ∩dµ)i+j, as desired. 2
By(♣),R(i)n is an ideal ofRn and, ifi1, thenR(i)n is a nilpotent ideal ofRn. Therefore,R(1)n ⊂RadRn. In fact:
(♦) RadRn= R(1)n .
Proof of(♦). First, note that RadRnconsists of the nilpotent elements ofRnbecauseRnis commutative. Now, letϕ be a nilpotent element ofRn. Writeϕ=
λ∈Part(n)aλϕ¯λand letλ0∈Part(n)be maximal (for the orderon Part(n)) such thataλ0=0. Then, by (6), the coefficient ofϕλ0 inϕr is equal toarλ
0|W (λ0)|r−1. Therefore, sinceϕis nilpotent andaλ0=0, we get thatpdivides|W (λ0)|, so thatλ0∈Part(p)1 (n)(by (3)). Consequently,ϕ−aλ0ϕ¯λ0 is nilpotent and we can repeat the argument to find finally thatϕ∈ R(1)n . 2
We shall now establish a consequence of (5) (or (6)). We need some notation. If α=(α1, . . . , αr) is a com- position of n and β =(β1, . . . , βs) is a composition of n, let αβ denote the composition of n+n equal to(α1, . . . , αr, β1, . . . , βr). If 1j n and if 0k[n/j], we denote byν(n, j, k) the composition(n−j k, j, j, . . . , j )ofn, wherej is repeatedktimes (ifn=j k, then the partn−j kis omitted). Ifλ∈Comp(n), we set
M(λ)= {0} ∪
1jn|pdoes not dividerj(λ) , m(λ)=maxM(λ),
J (λ)= {0} ∪
1jn|rj(λ)p , j(λ)=minJ (λ)
and
jm(λ)=
j(λ),m(λ) .
LetI= {0,1, . . . ,[n/p]}. Thenjm(λ)∈I×I. Let us now introduce an orderonI×I. If(j, m),(j, m)are two elements ofI×I, we write(j, m)(j, m)if one of the following two conditions is satisfied:
(a) j < j.
(b) j=jandmm.
Now, leti1 and letλ∈Part(p)i+1(n). Let(j, m)=jm(λ). Thenλ=αν0, whereαis a partition ofn−m−jp andν0=ν(m+jp, j, p). Letλ˜=α(m+jp). Thenπp(˜λ)=i(indeed,rm+jp(˜λ)=1+rm+jp(α)=1+rm+jp(λ) and, by the maximality ofm, we have thatpdividesrm+jp(λ)) and
(♥) ϕ¯ν(n,j,p)ϕ¯λ˜∈ ¯ϕλ+
µ∈Part(p)i+1(n) jm(µ)≺(j,m)
Fpϕ¯µ
.
Proof of(♥). Letν=ν(n, j, p). Sinceπp(λ)=i+12, there existsj∈ {1,2, . . . ,[n/p]}such thatrj(λ)p.
Then jj (by definition of j), so n2pj. In particular, n−pj > j, so W (ν)Sp. Now, if m> m, then rm(α)=rm(λ), so
(♥) ∀m> m, rm(α)≡0 modp.
Also
(♥) ∀l=m+jp, rl(˜λ)=rl(α) and
(♥) rm+jp(˜λ)=rm+jp(α)+1.
Now, keep the notation of (6). We may, and we will, assume that 1∈X
νλ˜. First, note thatν∩ ˜λ=αν0and that the image of∆˜1inW (ν)×W (˜λ)is equal toW (ν)×W (α). But, by(♥),(♥)and(♥), the index ofW (α)inW (˜λ)is
≡1 modp. Thus, by (6), we have
¯
ϕνϕ¯λ˜= ¯ϕλ+
d∈X
νλ˜−{1}
|W (ν)| · |W (λ)˜ |
|W (ν,λ, d)˜ | ϕ¯ν∩dλ˜.
Now, letd be an element ofXνλ˜ such thatpdoes not divide |W (ν)|W (ν,˜|·|W (λ)˜|
λ,d)| =xd and such thatjm(ν∩dλ)˜ jm(λ). It is sufficient to show thatd∈NνNλ˜. Writeα=(α1, . . . , αr). Then
ν∩dλ˜=(n1, . . . , nr, n0)j(1) · · · j(p),
where nk0 andj(l) is a composition of j with at most r+1 parts. Sincep does not divide xd, the image of Nν∩dNλ˜ inW (ν)contains a Sylowp-subgroup ofW (ν)Sp. Letw∈Nν∩dNλ˜be such that its image inW (ν)is an element of orderp. Then there existsσ∈Sν such thatwσ normalizesSν∩dλ˜. In particular,j(1)= · · · =j(p). So, ifj(1)=(j ), thenj(ν∩dλ) <˜ j(λ), which contradicts our hypothesis. Soj(1)= · · · =j(p)=(j ). Therefore,
λ˜∩d−1ν=ν(α1, j, k1) · · · ν(αr, j, kr)ν(m+jp, j, k0), where 0ki p andr
i=0ki =p. Note that (n1, . . . , nr, n0)=(α1−k1j, . . . , αr −krj, α0−k0j ) where, for simplification, we denote α0=m+jp. Also, j(˜λ∩d−1ν)j and, since jm(˜λ∩d−1ν)(j, m), we have that m(˜λ∩d−1ν)m. Recall thatd−1wd∈Nλ˜. So two cases may occur:
•Assume that there exists a sequence 0i1<· · ·< iprsuch that 0=ki1= · · · =kip(=1) and such thatαi1=
· · · =αip. Sorl(˜λ∩d−1ν)≡rl(˜λ)modpfor everyl1. In particular,rm+jp(˜λ∩d−1ν)≡1+rm+jp(α)≡1 modp by(♥)and(♥). Thus,m(˜λ∩d−1ν)m+jp > m, which contradicts our hypothesis.
•So, sinced−1wdbelongs toNλ˜and permutes thep=k0+k1+· · ·+krfactorsSjofd−1Sν, there exists a unique i∈ {0,1, . . . , r}such thatki=p. Consequently,ki =0 ifi=i. Ifαi > m+jp, thenrαi(˜λ∩d−1ν)=rαi(˜λ)−1= rαi(α)−1 (by(♥)), sopdoes not dividerαi(˜λ∩d−1ν)(by(♥)), which implies thatm(˜λ∩d−1ν)αi> m, contrarily to our hypothesis. Ifαi< m+jp, thenrm+jp(˜λ∩d−1ν)=rm+jp(˜λ)=rm+jp(α)+1 (by(♥)), sopdoes not divide rm+jp(˜λ∩d−1ν)(by(♥)), contrarily to our hypothesis. This shows thatαi=m+jp. In other words,d∈Nλ˜Nν, as desired. 2
By(♦), Theorem A follows immediately from the next result: ifi0, then (♠) R(1)n R(i)n = R(in+1).
Proof of (♠). We may assume thati1. By (♣), we have R(1)n R(i)n ⊂ R(in+1). So we only need to prove that, if λ∈Part(p)i+1(n)thenϕ¯λ∈ R(1)n R(i)n . But this follows from(♥)and an easy induction on jm(λ)∈I ×I (for the order). 2
References
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[2] M. Geck, G. Pfeiffer, Characters of Finite Coxeter Groups and Iwahori–Hecke Algebras, London Math. Soc. Monogr. (N.S.), vol. 21, The Clarendon Press, Oxford University Press, New York, 2000.