Semi-invariants of binary forms pertaining to a unimodality theorem of Reiner and Stanton

Abstract

The strange symmetric difference of the $q$-binomial coefficients $F_{n,k}(q)={n+k\brack k}-q^{n}{n+k-2\brack k-2}$ as called by Stanley, was introduced by Reiner and Stanton. They proved that $F_{n,k}(q)$ is symmetric and unimodal for $k\geq 2$ and any even nonnegative integer $n$ by using the representation theory for Lie algebras. Inspired by the connection between the Gaussian coefficients, or the $q$-binomial coefficients, and semi-invariants of binary forms established by Sylvester in his 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 $G_{n,k,r}(q)={n+k\brack k}-q^{nr/2}{n+k-r\brack k-r}$, where $n,r\geq8$, $k\geq r$ and either $n$ or $r$ is even.