论文标题
Petrie对称功能
Petrie symmetric functions
论文作者
论文摘要
对于任何积极的整数$ k $和非负整数$ m $，我们认为对称函数$ g \ weft（k，m \右）$定义为所有$ m $的单个单元$ $ m $的总和，仅涉及小于$ k $的指数。我们称$ g \ left（k，m \右）$是为了纪念弗林德斯·佩特里（Flinders Petrie）的“ petrie对称功能”，因为其在Schur基础上扩展的系数是Petrie矩阵的决定因素（因此属于$ \ weft \ left \ weft \ {0,1，-1，-1，-1，-1 \ right \} $ bicla a Classical classical classical of Gordon and Gordon和wilkins wilkins wilkins wilkins wilkins wilkins wilkins wilkins wilkins wilkins wilkins wilkins wilkins wilkins。更普遍地，我们证明了一个类似Pieri的规则，用于扩展$ g \ left（k，m \ right）\ cdots_μ$的产品，每当$μ$是分区时；此扩展中的所有系数都属于$ \ left \ {0,1，-1 \ right \} $。 We also show that $G\left( k,1\right) ,G\left( k,2\right) ,G\left( k,3\right) ,\ldots$ form an algebraically independent generating set for the symmetric functions when $1-k$ is invertible in the base ring, and we prove a conjecture of Liu and Polo about the expansion of $G\left( k,2k-1\right)$在Schur的基础上。
For any positive integer $k$ and nonnegative integer $m$, we consider the symmetric function $G\left( k,m\right)$ defined as the sum of all monomials of degree $m$ that involve only exponents smaller than $k$. We call $G\left( k,m\right)$ a "Petrie symmetric function" in honor of Flinders Petrie, as the coefficients in its expansion in the Schur basis are determinants of Petrie matrices (and thus belong to $\left\{ 0,1,-1\right\} $ by a classical result of Gordon and Wilkinson). More generally, we prove a Pieri-like rule for expanding a product of the form $G\left( k,m\right) \cdot s_μ$ in the Schur basis whenever $μ$ is a partition; all coefficients in this expansion belong to $\left\{ 0,1,-1\right\} $. We also show that $G\left( k,1\right) ,G\left( k,2\right) ,G\left( k,3\right) ,\ldots$ form an algebraically independent generating set for the symmetric functions when $1-k$ is invertible in the base ring, and we prove a conjecture of Liu and Polo about the expansion of $G\left( k,2k-1\right)$ in the Schur basis.
