Semi-invariants of Binary Forms Pertaining to a Unimodality Theorem of Reiner and Stanton

TL;DR

This paper explores the unimodality and strict unimodality of certain symmetric differences of q-binomial coefficients, providing new interpretations and extending previous results with combinatorial and algebraic insights.

Contribution

It offers a new interpretation of the unimodality of specific q-binomial differences using semi-invariants and proves their strict unimodality under broader conditions.

Findings

Proved symmetry and unimodality of F_{n,k}(q) for certain parameters.

Established strict unimodality of G_{n,k,r}(q) under specified conditions.

Connected unimodality properties to semi-invariants and representation theory.

Abstract

The symmetric difference of the $q$-binomial coefficients $F_{n,k}(q)={n+k\brack k}-q^{n}{n+k-2\brack k-2}$ was introduced by Reiner and Stanton. They proved that $F_{n,k}(q)$ is symmetric and unimodal for $k \geq 2$ and $n$ even by using the representation theory for Lie algebras. Based on Sylvester's proof of the unimodality of the Gaussian coefficients, as conjectured by Cayley, we find an interpretation of the unimodality of $F_{n,k}(q)$ in terms of semi-invariants. In the spirit of the strict unimodality of the Gaussian coefficients due to Pak and Panova, we prove the strict unimodality of the symmetric difference $G_{n,k,r}(q)={n+k\brack k}-q^{nr/2}{n+k-r\brack k-r}$, except for the two terms at both ends, where $n,r\geq8$, $k\geq r$ and at least one of $n$ and $r$ is even.