Coxeter群的扭曲对合的长度生成函数
Length-generating functions for twisted involutions of Coxeter groups
浏览论文内容
中文总结 AI 辅助
本文证明了有限秩Coxeter系统中扭曲对合的长度生成函数为有理函数，推导了类似抛物子群生长级数的递推关系，并推广到封闭子集，为Lusztig的幂级数恒等式提供了新证明。
中文摘要 AI 辅助
对于有限秩的Coxeter系统$(W,S)$以及一个保持$S$的对合自同构$\u2217:W\to W$，我们证明了$\u2217$-扭曲对合集合的长度生成函数$\sum_{z\in\mathbf{I}_{W,\ast}}q^{\ell(z)}$是有理函数。我们推导了与众所周知的递推关系类似的递推关系，这些递推关系用$W$的抛物子群的生长级数$W_I(q)$来表达生长级数$W(q):=\sum_{w\in W}q^{\ell(w)}$。我们的结果推广到任何在$\u2217$-扭曲共轭下封闭的$\u2217$-扭曲对合子集$\mathcal{C}$。我们利用这些递推关系给出了Lusztig关于扭曲对合的幂级数恒等式的另一种证明。
英文摘要
For a Coxeter system $(W,S)$ of finite rank and an involutive automorphism $\ast:W\to W$ which preserves $S$, we prove that the length-generating function $\sum_{z\in\mathbf{I}_{W,\ast}}q^{\ell(z)}$ of the set of $\ast$-twisted involutions is rational. We derive recurrence relations analogous to the well-known recurrence relations expressing the growth series $W(q):=\sum_{w\in W}q^{\ell(w)}$ in terms of the growth series $W_I(q)$ of the parabolic subgroups of $W$. Our result extends to any subset $\mathcal{C}$ of $\ast$-twisted involutions that is closed under $\ast$-twisted conjugation. We use these recurrence relations to give an alternative proof of a power-series identity for twisted involutions due to Lusztig.
