Poincar� polynomials of representations of finite groups generated by reflections

Molchanov, V. F.

doi:10.1007/bf01372350

Cited by 6 publications

(8 citation statements)

References 4 publications

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“…This conjecture was implicitly proved in type A n comparing the "First Layer Formulae" of Stembridge [21] with the results due to Kirillov, Pak [13] and Molchanov [16]. Moreover some general formulae for graded multiplicities are proved when V λ is the adjoint or the little adjoint module in [2,[7][8][9] and [22].…”

Section: Introductionmentioning

confidence: 93%

Reeder’s Conjecture for Even Orthogonal Lie Algebras

Trani

2022

Algebr Represent Theor

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Section: Introductionmentioning

confidence: 93%

Reeder’s Conjecture for Even Orthogonal Lie Algebras

Trani

2022

Algebr Represent Theor

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“…This theorem is [40, exercise 7.50] and the solution utilizes [40,Theorem 7.21.2], which is noted to have first been explicitly stated by Stanley in [39]. The theorem also appeared in [28], which provides a similar factorization for the R * λ µ (t) for the real reflection group {±1} n ⋊ S n of type B n , a sum formula for the analogous R * χ (t) for the real reflection group of type D n , and a factorization result for the R * χ (t) for dihedral groups. Renteln [29] uses Theorem 6.2 and Equation (6.1) to provide a combinatorial formula for the codimension (equivalently distance) spectra in type A, leaves similar formulas in types B and D to the reader, and includes the spectra for dihedral type.…”

Section: Poincaré Polynomialsmentioning

confidence: 99%

“…. It is then possible to use Identity (6.2), along with properties of determinants and the factorization of R * triv (t) from Example 6.1, to determine the roots of s λ (t) and R * λ (t) (see [28]).…”

Section: Substitutionmentioning

confidence: 99%

“…When p = 1, the Poincaré polynomials R * χ (t) for G(r, p, n) do not always have all integer roots. This is perhaps not too surprising since in the case r = 2, for example, Molchanov [28] expresses the Poincaré polynomials of the groups of type D n as averages of those of the groups of type B n . We suspect consideration of the Clifford theory relating characters of G(r, p, n) to those of G(r, 1, n) might allow for generally expressing the Poincaré polynomials of G(r, p, n) as averages of those of G(r, 1, n).…”

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confidence: 99%

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Spectra of Cayley graphs of complex reflection groups

Foster-Greenwood

Kriloff

2015

J Algebr Comb

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ABSTRACT. Renteln proved that the eigenvalues of the distance matrix of a Cayley graph of a real reflection group with respect to the set of all reflections are integral and provided a combinatorial formula for some such spectra. We prove the eigenvalues of the distance, adjacency, and codimension matrices of Cayley graphs of complex reflection groups with connection sets consisting of all reflections are integral and provide a combinatorial formula for the codimension spectra for a family of monomial complex reflection groups.

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“…o (4.11) Remark. If G is a Coxeter group of type A I or B I the polynomials St have integer roots for all irreducible characters t. This was proved by D. E. Littlewood and A. R. Richardson [3, p. 56] in the case of Al and by V. F. Molchanov [4] and W. Ostertag [8] in the case of B I • In both cases there is a pleasant formula for the roots of St in terms of Young diagrams.…”

Section: Let D=d(g) Be the Coxeter Arrangement And Let He D Thenmentioning

confidence: 94%

On Coxeter Arrangements and the Coxeter Number

Orlik¹,

Solomon²,

Terao³

Advanced Studies in Pure Mathematics

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Let (G, V) be an irreducible Coxeter group and let d be the corresponding Coxeter arrangement. Let HE d be a hyperplane and let d H be the restriction of d to H. Let h be the Coxeter number. We prove that and show that d H is a free arrangement whose degrees are ml , •• " m l _ 1 the first I-I exponents G. § 1. Introduction Let V be Euclidean space of dimension I with positive definite inner product written ( ,). A hyperplane in V is a vector subspace of codimension 1.Let S=S(V*) be the symmetric algebra of the dual space V*. We may view S as the graded algebra of polynomial functions on V. Let Der(S) be the S-module of all derivations of S. A non-zero element () E Der(S) is said to be homogeneous of degree b, written deg (}=b, if (}(Sp)c Sp+b' This makes Der(S) a graded S-module. Choose, for each HE d, a linear form a H E V* with ker(aH)=H. Define Q E S by (1.1) The polynomial Q is uniquely determined by d up to a constant multiple. Define (1.2) D(d)={(} E Der(S)I(}Q E QS}.Then D(d) is a graded S-submodule of Der(S).(1.3) Definition [14]. The arrangement d is free if D(d) is a free S-module.

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Poincar� polynomials of representations of finite groups generated by reflections

Cited by 6 publications

References 4 publications

Reeder’s Conjecture for Even Orthogonal Lie Algebras

Reeder’s Conjecture for Even Orthogonal Lie Algebras

Spectra of Cayley graphs of complex reflection groups

On Coxeter Arrangements and the Coxeter Number

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