A diagrammatic approach to Kronecker squares

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:127

Article

Vallejo, Ernesto

关键词: Kronecker product;    Young tableau;    Schur function;    Kostka number;    Littlewood-Richardson rule;   

DOI  :  10.1016/j.jcta.2014.06.002

来源: Elsevier

【 摘 要 】

In this paper we improve a method of Robinson and Taulbee for computing Kronecker coefficients and show that for any partition (nu) over bar of d there is a polynomial k((nu) over bar), with rational coefficients in variables x(C), where C runs over the set of isomorphism classes of connected skew diagrams of size at most d, such that for all partitions lambda of n, the Kronecker coefficient g(lambda, lambda, (n - d, (nu) over bar)) is obtained from k((nu) over bar)(x(C)) substituting each x(C) by the number of partitions a contained in lambda such that lambda/alpha is in the class C. Some results of our method extend to arbitrary Kronecker coefficients. We present two applications. The first is a contribution to the Saxl conjecture, which asserts that if rho(k) = (k,k - 1,...,2,1) is the staircase partition, then the Kronecker square chi(rho) circle times chi(rho) contains every irreducible character of the symmetric group as a component. Here we prove that for any partition (nu) over bar of size d there is a piecewise polynomial function s(<(nu)over bar) in one real variable such that for all k, one has g(rho k, rho k, (vertical bar rho k vertical bar - d, <(nu)over bar>)) = S-(nu) over bar(k). The second application is a proof of a new stability property for arbitrary Kronecker coefficients. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free