Bounds on Kronecker and $q$-binomial coefficients

TL;DR

This paper establishes new bounds on Kronecker and q-binomial coefficients, extending classical results and providing explicit estimates and asymptotic bounds for these combinatorial quantities.

Contribution

It introduces a lower bound on Kronecker coefficients using symmetric group characters and extends Sylvester's unimodality to derive sharp bounds on q-binomial coefficient differences.

Findings

Derived explicit estimates for Kronecker coefficients.

Extended Sylvester's unimodality to q-binomial coefficients.

Provided asymptotic lower bounds for a broader class of Kronecker 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 $\binom{n}{k}_q$ 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.