Unimodality via Kronecker products

TL;DR

This paper introduces algebraic proofs and generalizations for the unimodality of q-binomial coefficients using Kronecker coefficients, extending to strict unimodality and partition statistics.

Contribution

It provides new algebraic proofs and broader generalizations of unimodality for q-binomial coefficients via Kronecker coefficients.

Findings

Proves unimodality of q-binomial coefficients using Kronecker coefficients.

Establishes strict unimodality of diagonal q-binomial coefficients.

Shows unimodality of certain partition statistics.

Abstract

We present new proofs and generalizations of unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker coefficients of representations of S_n. Other applications of this approach include strict unimodality of the diagonal q-binomial coefficients and unimodality of certain partition statistics.