Horn inequalities for nonzero Kronecker coefficients

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In p ro g re ss Horn inequalities for nonzero Kronecker coefficients N. Ressayre May 25, 2011 Abstract 1 Introduction Let ? = (?1 ≥ ?2 ≥ · · · ≥ ?e ≥ 0) be a partition. Set |?| = ∑ i ?i. Then ? is a partition of |?|. Consider the symmetric group Sn acting on n letters. The irreducible representations of Sn are parametrized by the partitions of n, see e.g. [Mac95, I. 7] . Let [?] denote the representation corresponding to ?. The Kronecker coefficients k?? ? , depending on three partitions ?? and ? of the same integer n, are defined by the identity [?]? [?] = ∑ ? k?? ? [?]. (1) A classical result due to Murnaghan is the following statement. Proposition 1 If k?? ? 6= 0 then we have (n? ?1) + (n? ?1) ≥ n? ?1. (2) The aim of this note is to prove many other explicit inequalities which are consequences of the nonvanishing of a Kronecker coefficient. The length l(?) of the partition ? is the number of nonzero parts ?i. Let V be a complex vector space of dimension d. If l(?) ≤ d then S?V denotes the Schur power (see e.

c???s ?v

using schur-weyl

kronecker coefficient

littlewood-richardson coefficient

f0 ?

group acting

semigroups

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Publié par

profil-urra-2012

Langue

English

Littlewood?Richardson rule Semigroup

Horn inequalities for nonzero Kronecker coeﬃcients

N. Ressayre

May 25, 2011

Abstract

1 Introduction P Letα= (α1≥α2≥ ∙ ∙ ∙ ≥αe≥0) be a partition. Set|α|=αi. Thenα i is a partition of|α|the symmetric group. Consider Snacting onnletters. The irreducible representations ofSnare parametrized by the partitions of n, seee.g.[Mac95, I. 7] . Let [α] denote the representation corresponding toαKronecker coeﬃcients. The kα β γ, depending on three partitionsα β andγof the same integern, are deﬁned by the identity X [α]⊗[β] =kα β γ[γ].(1) γ

A classical result due to Murnaghan is the following statement.

Proposition 1Ifkα β γ6= 0then we have

(n−α1) + (n−β1)≥n−γ1.

(2)

The aim of this note is to prove many other explicit inequalities which are consequences of the nonvanishing of a Kronecker coeﬃcient.

The lengthl(α) of the partitionαis the number of nonzero partsαi. α LetVbe a complex vector space of dimensiond. Ifl(α)≤dthenS V denotes the Schur power (seee.g.is an irreducible polynomial[Ful91]): it representation of the linear group GL(V).

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