Kronecker Coefficients For Some Near-Rectangular Partitions

We give formulae for computing Kronecker coefficients occurring in the expansion of $s_{\mu}*s_{\nu}$, where both $\mu$ and $\nu$ are nearly rectangular, and have smallest parts equal to either 1 or 2. In particular, we study $s_{(n,n-1,1)}*s_{(n,n)}$, $s_{(n-1,n-1,1)}*s_{(n,n-1)}$, $s_{(n-1,n-1,2)}*s_{(n,n)}$, $s_{(n-1,n-1,1,1)}*s_{(n,n)}$ and $s_{(n,n,1)}*s_{(n,n,1)}$. Our approach relies on the interplay between manipulation of symmetric functions and the representation theory of the symmetric group, mainly employing the Pieri rule and a useful identity of Littlewood. As a consequence of these formulae, we also derive an expression enumerating certain standard Young tableaux of bounded height, in terms of the Motzkin and Catalan numbers

The stability of the Kronecker products of Schur functions

In the late 1930's Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n for which all the coefficients of a Kronecker product of Schur functions stabilize. We also compute two new bounds for the stabilization of a sequence of coefficients and show that they improve existing bounds of M. Brion and E. Vallejo.Comment: 16 page

A diagrammatic approach to Kronecker squares

In this paper we apply a method of Robinson and Taulbee for computing Kronecker coefficients together with other ingredients and show that the multiplicity of each component in a Kronecker square can be obtained from an evaluation of a certain polynomial, which depends only on the component and is computed combinatorially. This polynomial has as many variables as the set of isomorphism classes of connected skew diagrams of size at most the depth of the component. We present two applications. The first is a contribution to Saxl conjecture, which asserts that the Kronecker square of the staircase partition, contains every irreducible character of the symmetric group as a component. We prove that for any partition there is a piecewise polynomial function in one real variable such that for all k, such that the multiplicity of this partition in the Kronecker square of the staircase partition of size k is given by the evaluation of the polynomial function in k. The second application is a proof of a new stability property for Kronecker coefficients.Comment: 40 pages, 1 table, second version that incorporates reviewers suggestions, accepted for publication in Journal of Combinatorial Theory

Positivity of the symmetric group characters is as hard as the polynomial time hierarchy

We prove that deciding the vanishing of the character of the symmetric group is $C_=P$-complete. We use this hardness result to prove that the the square of the character is not contained in $\#P$, unless the polynomial hierarchy collapses to the second level. This rules out the existence of any (unsigned) combinatorial description for the square of the characters. As a byproduct of our proof we conclude that deciding positivity of the character is $PP$-complete under many-one reductions, and hence $PH$-hard under Turing-reductions.Comment: 15 pages, 1 figur