非零高阶Specht多项式及三行与钩形Garsia-Procesi模的构造

中文摘要 AI 辅助

给定一个多项式环商$R=\mathbb{C}[x_1,\ldots,x_n]/I$，其具有由变量置换诱导的对称群作用，高阶Specht基是其分解中每个不可约$\mathfrak{S}_n$-模的一组基的集合，这些基在最低次数中模仿经典Specht多项式构造的行为。目前，高阶Specht基已为余不变环、完整多项式环、$t=0$ Delta猜想中出现的环$R_{n,k}$、钩形Garsia-Haiman模以及两行Garsia-Procesi模等构造出来。我们建立了一个一般理论来确定高阶Specht多项式何时非零，并针对三行形状和钩形形状的所有Garsia-Procesi模，给出了一个猜想的高阶Specht基的证明。

英文摘要

Given a polynomial ring quotient $R=\mathbb{C}[x_1,\ldots,x_n]/I$ with an action of the symmetric group induced by permuting the variables, a higher Specht basis is a collection of bases for each irreducible $\mathfrak{S}_n$-module in its decomposition that mimics the behavior of the classical Specht polynomial construction in the lowest degrees. Higher Specht bases have now been constructed for the coinvariant ring, the full polynomial ring, the rings $R_{n,k}$ appearing in the $t=0$ Delta conjecture, the hook shape Garsia-Haiman modules, and the two-row Garsia-Procesi modules, and more. We establish a general theory for determining when a higher Specht polynomial is nonzero, and give a proof of a conjectural higher Specht basis for all Garsia-Procesi modules in the cases of three row shapes and hook shapes.

发表机构

• UC San Diego（加州大学圣迭戈分校）
• Colorado State University（科罗拉多州立大学）
• University of Pennsylvania（宾夕法尼亚大学）

机构由 AI 辅助整理，请以论文原文为准。