arXiv 1004.40132010-04math.RTmath.GR

上同调增长率与Kazhdan-Lusztig多项式

Cohomological growth rates and Kazhdan-Lusztig polynomials

Brian Parshall, Leonard Scott

备注：: 21 pages, 1 figure. This is a revision (the title has changed, and new results on rates of growth for $G$-modules are obtained). In third edition, some minor mistakes have been corrected

摘要：

在以前的工作中，作者建立了半单代数群$G$（正特征$p$）和（Lusztig）量子群$U_ζ$（单位$ζ$的根）的不可约模的次数$n$上同调和$\Ext$ -群的维数的各种界限.这些界限只取决于根系，而不是在特征$p$或单位$ζ$的根的大小。本文研究了这些界限的增长率。无论是在量子和代数群的情况下，这些增长率代表新的和基本的不变量附加到根系统$Φ$。对于具有固定$Φ$的量子群$U_ζ$，证明了序列$\{\max_{L\,{\text{irred}}}\dim \opH^n(U_ζ,L)\}_n$具有与$ζ$无关的多项式增长.事实上，我们提供了多项式增长率的上界和下界。应用这些和相关的结果$\Ext^n_{U_ζ}$的Kazhdan-Lusztig多项式。多项式增长在代数群的情况下仍然是一个悬而未决的问题，虽然它被证明，$\{\log\max_{L\,\text{\rm irred}}\dim\opH^n(G,L)\}$有多项式增长$\leq 3$任何固定的素数$p$（和$\leq 4$，如果$p$是允许的变化与$n$）。我们表明这些问题的相关性（额外的结构）的常数提出的理论中的高上同调群有限简单的组与不可约系数的Guralnick，坎特，Kassabov，和Lubotzky \cite{GKKL}。

英文摘要：

In previous work, the authors established various bounds for the dimensions of degree $n$ cohomology and $\Ext$-groups, for irreducible modules of semisimple algebraic groups $G$ (in positive characteristic $p$) and (Lusztig) quantum groups $U_ζ$ (at roots of unity $ζ$). These bounds depend only on the root system, and not on the characteristic $p$ or the size of the root of unity $ζ$. This paper investigates the rate of growth of these bounds. Both in the quantum and algebraic group situation, these rates of growth represent new and fundamental invariants attached to the root system $Φ$. For quantum groups $U_ζ$ with a fixed $Φ$, we show the sequence $\{\max_{L\,{\text{irred}}}\dim \opH^n(U_ζ,L)\}_n$ has polynomial growth independent of $ζ$. In fact, we provide upper and lower bounds for the polynomial growth rate. Applications of these and related results for $\Ext^n_{U_ζ}$ are given to Kazhdan-Lusztig polynomials. Polynomial growth in the algebraic group case remains an open question, though it is proved that $\{\log\max_{L\,\text{\rm irred}}\dim\opH^n(G,L)\}$ has polynomial growth $\leq 3$ for any fixed prime $p$ (and $\leq 4$ if $p$ is allowed to vary with $n$). We indicate the relevance of these issues to (additional structure for) the constants proposed in the theory of higher cohomology groups for finite simple groups with irreducible coefficients by Guralnick, Kantor, Kassabov, and Lubotzky \cite{GKKL}.

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。

摘要：

英文摘要：

相关论文