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

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.