多对称函数的转移矩阵与Pieri型规则
Transition matrices and Pieri-type rules for polysymmetric functions
AI总结：
本文针对Asvin G和Andrew O'Desky提出的多对称函数分次代数PΛ，推导了其乘法规则与转移矩阵项的组合公式，给出非纯基在纯张量基中的展开式，涉及推广的类表格结构及新Pieri型规则。
摘要：
Asvin G和Andrew O'Desky最近引入了多对称函数的分次代数PΛ，作为对称函数代数Λ的推广。本文推导了PΛ的一些乘法规则和转移矩阵项的组合公式，这些公式类似于Λ中著名的经典公式。具体而言，我们考虑PΛ的纯张量基{s^⊗_τ}、{p^⊗_τ}和{m^⊗_τ}，它们是Λ的经典Schur基、幂和基和单项式基的张量积。我们找到了Asvin G和O'Desky研究的非纯基{P_δ}、{H_δ}、{E^+_δ}和{E_δ}在这些基中的展开式。答案涉及推广半标准表格、边缘钩表格以及Eğecioğlu和Remmel的砖块表格的类表格结构。这些对象通过新的Pieri型规则的迭代产生，该规则给出了如s^⊗_σH_δ、p^⊗_σE_δ等乘积的展开式。
英文摘要：
Asvin G and Andrew O'Desky recently introduced the graded algebra P$Λ$ of polysymmetric functions as a generalization of the algebra $Λ$ of symmetric functions. This article develops combinatorial formulas for some multiplication rules and transition matrix entries for P$Λ$ that are analogous to well-known classical formulas for $Λ$. In more detail, we consider pure tensor bases $\{s^{\otimes}_τ\}$, $\{p^{\otimes}_τ\}$, and $\{m^{\otimes}_τ\}$ for P$Λ$ that arise as tensor products of the classical Schur basis, power-sum basis, and monomial basis for $Λ$. We find expansions in these bases of the non-pure bases $\{P_δ\}$, $\{H_δ\}$, $\{E^+_δ\}$, and $\{E_δ\}$ studied by Asvin G and O'Desky. The answers involve tableau-like structures generalizing semistandard tableaux, rim-hook tableaux, and the brick tabloids of Eğecioğlu and Remmel. These objects arise by iteration of new Pieri-type rules that give expansions of products such as $s^{\otimes}_σH_δ$, $p^{\otimes}_σE_δ$, etc.
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