Coxeter多项式的Chebycheff递推公式

Coxeter combinatorics for sum formulas in the representation theory of algebraic groups

Let $G$ be a simple algebraic group over an algebraically closed field $\mathbb{F}$ of characteristic $p\geq h$, the Coxeter number of $G$. We observe an easy `recursion formula' for computing the Jantzen sum formula of a Weyl module with $p$-regular highest weight. We also discuss a `duality formula' that relates the Jantzen sum formula to Andersen's sum formula for tilting filtrations and we give two different representation theoretic explanations of the recursion formula. As a corollary, we also obtain an upper bound on the length of the Jantzen filtration of a Weyl module with $p$-regular …

Parabolic recursions for Kazhdan-Lusztig polynomials and the hypercube decomposition

We employ general parabolic recursion methods to demonstrate the recently devised hypercube formula for Kazhdan-Lusztig polynomials of $S_n$, and establish its generalization to the full setting of a finite Coxeter system through algebraic proof. We introduce procedures for positive decompositions of $q$-derived Kazhdan-Lusztig polynomials within this setting, that utilize classical Hecke algebra positivity phenomena of Dyer-Lehrer and Grojnowski-Haiman. This leads to a distinct algorithmic approach to the subject, based on induction from a parabolic subgroup. We propose suitable weak variants…

Kazhdan-Lusztig理论中Deodhar元素的嵌入因子模式

Embedded factor patterns for Deodhar elements in Kazhdan-Lusztig theory

有限外尔群的Kazhdan-Lusztig多项式出现在舒伯特簇的几何和表示论中。它被证明后不久，他们的介绍，他们有非负整数系数，但没有完全的组合解释，他们是众所周知的一般。Deodhar（1990）给出了计算Kazhdan-Lusztig多项式的框架，其中通常涉及递归。我们定义嵌入式因子模式避免一般Coxeter群和使用它来表征时Deodhar的算法产生一个简单的组合公式的Kazhdan-Lusztig多项式的有限外尔群。等价地，如果$(W, S)$是有限外尔群的Coxeter系统，则我们对Kazhdan-Lusztig基元$C'_w$可以写成$C'_s$的单项式的元$w \in W$进行分类，其中$s \in S$。这一工作推广了Billey-Warrington（2001）和Fan-Green（1997）的结果，前者将$A$型中的Deodhar元素确定为321-六边形回避排列，后者确定了全紧Coxeter群。