The symmetric difference of the q-binomial coefficients Fn,k(q) = n+k k − qnn+k−2 k−2 was introduced by Reiner and Stanton. They proved that Fn,k(q) is symmetric and unimodal for k ≥ 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 Fn,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 Gn,k,r(q) = n+k k − qnr/2n+k−r k−r, except for the two terms at both ends, where n,r ≥ 8, k ≥ r and at least one of n and r is even.

中文翻译：

---

与 Reiner 和 Stanton 的单峰定理有关的二元形式的半不变量

对称差q-二项式系数Fn,ķ(q) = n+ķ ķ - qnn+ķ-2 ķ-2由 Reiner 和 Stanton 介绍。他们证明了Fn,ķ(q)是对称且单峰的ķ ≥ 2和n甚至通过使用李代数的表示论。基于西尔维斯特对高斯系数单峰性的证明，正如 Cayley 所推测的，我们找到了对单峰性的解释Fn,ķ(q)就半不变量而言。本着由于 Pak 和 Panova 导致的高斯系数的严格单峰性的精神，我们证明了对称差的严格单峰性Gn,ķ,r(q) = n+ķ ķ - qnr/2n+ķ-r ķ-r，除了两端的两项，其中n,r ≥ 8,ķ ≥ r和至少其中之一n和r甚至。