Lie模及其复杂性
The Lie module and its complexity
摘要：
模的复杂度是一个重要的同调不变量，它度量了模的最小投射分解的多项式增长率。对于对称群$Σ_n$，李模$\mathsf{Lie}(n)$近年来引起了人们极大的兴趣.本文证明了特征$p$中的$\mathsf{Lie}(n)$的复杂度是$t$，其中$p^t$是$p$除$n$的最大幂，从而证明了Erdmann，Lim和Tan的一个猜想.证明使用的工作Arone和Kankaanrinta其中描述了同源性$\operatorname{H}_\bullet(Σ_n, \mathsf{Lie}(n))$和早期的工作Hemmer和中野的复杂性模块超过$Σ_n$，涉及限制到年轻的子群。
英文摘要：
The complexity of a module is an important homological invariant that measures the polynomial rate of growth of its minimal projective resolution. For the symmetric group $Σ_n$, the Lie module $\mathsf{Lie}(n)$ has attracted a great deal of interest in recent years. We prove here that the complexity of $\mathsf{Lie}(n)$ in characteristic $p$ is $t$ where $p^t$ is the largest power of $p$ dividing $n$, thus proving a conjecture of Erdmann, Lim and Tan. The proof uses work of Arone and Kankaanrinta which describes the homology $\operatorname{H}_\bullet(Σ_n, \mathsf{Lie}(n))$ and earlier work of Hemmer and Nakano on complexity for modules over $Σ_n$ that involves restriction to Young subgroups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
