Bounds on Kronecker and $q$-binomial coefficients

Abstract

We present a lower bound on the Kronecker coefficients for tensor squares of the symmetric group via the characters of~$S_n$, which we apply to obtain various explicit estimates. Notably, we extend Sylvester's unimodality of $q$-binomial coefficients ${\left(\genfrac{}{}{0ex}{}{n}{k}\right)}_q$ as polynomials in~$q$ to derive sharp bounds on the differences of their consecutive coefficients. We then derive effective asymptotic lower bounds for a wider class of Kronecker coefficients.

version May 2016: improved the effective constants. To appear in JCTA. This paper is an extension of parts of the earlier paper "Bounds on the Kronecker coefficients" arXiv:1406.2988, which also contains stability results

References in corpus (4)

• On the complexity of computing Kronecker coefficients
• Bounds on the Kronecker coefficients
• Enumeration of Standard Young Tableaux
• A combinatorial proof of strict unimodality for $q$-binomial coefficients

Cited by in corpus (1)

• Skew cell modules for diagram algebras