Representation theory and homological stability
Representation theory and homological stability
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DOI：
10.1016/j.aim.2013.06.016
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发表时间：
2010-08
影响因子：
1.7
通讯作者：
Thomas Church;B. Farb
中科院分区：
文献类型：
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作者：
Thomas Church;B. Farb
我们引入了组 G n 的表示序列 V n 的表示稳定性（和几种变体）的思想。我们在这里介绍的新观点的一个核心应用是将表示论引入同调稳定性的研究中。这使得将同调稳定性的经典定理扩展到更广泛的例子成为可能。表示稳定性还提供了一个寻找和预测模式的框架，从经典表示理论（Littlewood–Richardson 和 Murnaghan 规则、Schur 函子的稳定性）到群的上同调（纯辫子群、Torelli 群和同余群）、李代数及其同源性、flag 和 Schubert 簇的（等变）上同调、组合数学（(n+1) n−1 猜想）。本文的大部分内容致力于通过示例来揭示这一现象。通过这样做，我们获得了应用、定理和猜想。除了发现新现象之外，表征稳定性的观点还可用于解决理论之外的问题。除了本文给出的应用之外，Church-Ellenberg-Farb（准备中）[20]也将其应用于数论和有限群论中的计数问题。 Church (2012)[19] 也使用表示稳定性来给出有向流形上配置空间的经典同调稳定性定理的广泛概括和新证明。
We introduce the idea of representation stability (and several variations) for a sequence of representations V n of groups G n. A central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability. This makes it possible to extend classical theorems of homological stability to a much broader variety of examples. Representation stability also provides a framework in which to find and to predict patterns, from classical representation theory (Littlewood–Richardson and Murnaghan rules, stability of Schur functors), to cohomology of groups (pure braid, Torelli and congruence groups), to Lie algebras and their homology, to the (equivariant) cohomology of flag and Schubert varieties, to combinatorics (the (n+ 1) n− 1 conjecture). The majority of this paper is devoted to exposing this phenomenon through examples. In doing this we obtain applications, theorems and conjectures. Beyond the discovery of new phenomena, the viewpoint of representation stability can be useful in solving problems outside the theory. In addition to the applications given in this paper, it is applied by Church–Ellenberg–Farb (in preparation)[20] to counting problems in number theory and finite group theory. Representation stability is also used by Church (2012)[19] to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds.