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TENSORIAL SQUARE OF THE HYPEROCTAHEDRAL GROUP COINVARIANT SPACE

28 pages
TENSORIAL SQUARE OF THE HYPEROCTAHEDRAL GROUP COINVARIANT SPACE FRANC¸OIS BERGERON AND RICCARDO BIAGIOLI Abstract. The purpose of this paper is to give an explicit description of the trivial and alternating components of the irreducible representation decomposition of the bigraded module obtained as the tensor square of the coinvariant space for hyperoctahedral groups. Contents 1. Introduction. 1 2. Reflection group action on the polynomial ring 2 3. Diagonally invariant and alternating polynomials 5 4. The Hyperoctahedral group Bn 7 5. Plethystic substitution 9 6. Frobenius characteristic of Bn-modules 11 7. Frobenius characteristic of Q[x]. 13 8. Frobenius characteristic of H. 13 9. The trivial component of C 15 10. Combinatorics of e-diagrams 18 11. Compactification of e-diagrams 21 12. The bijection 22 13. A basis for the trivial component of C 23 14. The alternating component of C 24 15. Compact o-diagrams 26 References 28 1. Introduction. The group Bn ? Bn, with Bn the group of “signed” permutations, acts in the usual natural way as a “reflection group” on the polynomial ring in two sets of variables Q[x,y] = Q[x1, . . . , xn, y1, . . . , yn], (?, ?) (xi, yj) = (±x?(i),±y?(j)), where ± denotes some appropriate sign, and ? and ? are the unsigned permutations cor- responding respectively to ? and ?.

  • diagonally invariant

  • polynomial ring

  • finite reflection

  • group

  • qw

  • diagonal bn

  • invariant subspaces

  • reflection group


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TENSORIAL SQUARE OF THE HYPEROCTAHEDRAL GROUP COINVARIANT SPACE
FRAN¸COISBERGERONANDRICCARDOBIAGIOLI
Abstract.The purpose of this paper is to give an explicit description of the trivial and alternating components of the irreducible representation decomposition of the bigraded module obtained as the tensor square of the coinvariant space for hyperoctahedral groups.
Contents
1. Introduction. 2. Reflection group action on the polynomial ring 3. Diagonally invariant and alternating polynomials 4. The Hyperoctahedral groupBn 5. Plethystic substitution 6. Frobenius characteristic ofBn-modules 7. Frobenius characteristic ofQ[x]. 8. Frobenius characteristic ofH. 9. The trivial component ofC 10. Combinatorics ofesragimda-11. Compactification ofe-diasmarg 12. The bijection 13. A basis for the trivial component ofC 14. The alternating component ofC 15. Compacto-dsamgria References
1.Iductntroion.
1 2 5 7 9 11 13 13 15 18 21 22 23 24 26 28
The groupBn×Bn, withBnthe group of “signed” permutations, acts in the usual natural way as a “reflection group” on the polynomial ring in two sets of variables
Q[x,y] =Q[x1, . . . , xn, y1, . . . , yn],(β, γ) (xi, yj) = (±xσ(i),±yτ(j)), where±denotes some appropriate sign, andσandτare the unsigned permutations cor-responding respectively toβandγ. Let us denoteIB×Bthe ideal generated by constant term free invariant polynomials for this action. We then consider the “coinvariant space” C=Q[x,y]IB×Bfor the groupBn×Bn is well known [12, 13, 17, 18] that. ItCis iso-morphic to the regular representation ofBn×Bn, since it acts here as a reflection group.
Date: May 6, 2005. F.BergeronissupportedinpartbyNSERC-CanadaandFQRNT-Qu´ebec. 1
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F. BERGERON AND R. BIAGIOLI
The purpose of this paper is to study the isotypic components ofCwith respect to the action ofBnrather then that ofBn×Bn is to say that we are restricting the action. This toBn, here considered as a “diagonal” subgroup ofBn×Bn is worth underlying that. It this is not a reflection groups action, and so we are truly in front of a new situation with respect to the classical results alluded to above. Part of the results we obtain consists in giving explicit descriptions of the trivial and alternating components of the spaceC. We will see that in doing so, we are lead to introduce two new classes of combinatorial objects respectfully calledcompacte-diagramsandcompacto-diagrams. We give a nice bijection between naturaln-element subsets ofN×Nindexing “diagonalBn-invariants”, and triples (Dβ, λ, µ), whereλandµare partitions with at mostnparts, andDβis a compacte-diagram. A similar result will be obtained for “diagonalBn-alternants”, involving compact o we will also see, both families of compact diagrams are naturally As-diagram in this case. indexed by signed permutations.
One of the many reasons to study the coinvariant spaceCis that it strictly contains the space of “diagonal coinvariants” ofBnin a very natural way. will be further discussed This in Section 3. This space of diagonal coinvariants was recently characterized by Gordon [9] in his solution of conjectures of Haiman [11], (see also, [4], [10], and [14]). Moreover, the spaceCcontains some of the spaces that appear in the work of Allen [2], namely those that are generated byBn-alternants that are contained inC our Theorem 15.2 one can. Using readily check that the join of these spaces is strictly contained as a subspace ofC.
The paper is organized as follows. We start with a general survey of classical results regarding coinvariant spaces of finite reflection groups, followed with implications regarding the tensor square of these same spaces. We then specialize our discussion to hyperoctahedral groups, recalling in the process the main aspects of their representation theory relating to theBn of these results have already appeared Many-Frobenius transform of characters. in scattered publications but are hard to find in a unified presentation. We then finally proceed to introduce our combinatorial tools and derive our main results.
2.Reflection group action on the polynomial ring
For any finite reflection groupW, on a finite dimensional vector spaceVoverQ, there corresponds a natural action ofWon the polynomial ringQ[V]. In particular, ifx= x1, . . . , xnis a basis ofVthenQ[V] can be identified with the ringQ[x] of polynomials in the variablesx1, . . . , xn. As usual, we denote wp(x) =p(wx),
the action in question. It is clear that this action ofWis degree preserving, thus making natural the following considerations. Let us denoteπd(p(x)) the degreedhomogeneous component of a polynomialp(x ring). TheQ:=Q[x] is graded by degree, hence Q'MQd, d0
whereQd:=πd(Q) is thedegreedhomogeneous componentofQ that a subspace. RecallS is said to beegenohomuosifπd(S)Sfor alld. Whenever this is the case, we clearly have S=Ld0Sd, withSd:=SQd, and thus it makes sense to consider theHilbert seriesof