论文标题
二进制形式的半不变，与Reiner和Stanton的单峰定理有关
Semi-invariants of Binary Forms Pertaining to a Unimodality Theorem of Reiner and Stanton
论文作者
论文摘要
$ q $ biNamial系数的对称差异$ f_ {n，k}（q）= {n+k \ brack k} -q^{n} {n+k-2 \ brack k-2} $由莱因纳和斯坦顿引入。他们证明$ f_ {n，k}（q）$是对称的，对于$ k \ geq 2 $和$ n $，即使是通过使用代表理论来代数为lie代数。根据凯利（Cayley）的猜想，基于西尔维斯特（Sylvester）对高斯系数的单程性的证明，我们在半不变方面发现了$ f_ {n，k}（q）$的单型的解释。本着高斯和帕诺瓦引起的高斯系数的严格不像的精神，我们证明了对称差异的严格毫无印度，$ g_ {n，k，r}（q）= {n+k \ brack k} -q} -Q^Q^nr/2} $ n，r \ geq8 $，$ k \ geq r $和$ n $和$ r $的至少一个均匀。
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.
