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

Abstract

The symmetric difference of the [Formula: see text]-binomial coefficients [Formula: see text] was introduced by Reiner and Stanton. They proved that [Formula: see text] is symmetric and unimodal for [Formula: see text] and [Formula: see text] 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 [Formula: see text] 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 [Formula: see text], except for the two terms at both ends, where [Formula: see text], [Formula: see text] and at least one of [Formula: see text] and [Formula: see text] is even.