FI-modules and stability for representations of symmetric groups
FI-modules and stability for representations of symmetric groups
复制标题
DOI：
10.1215/00127094-3120274
复制
发表时间：
2012-04
期刊：
影响因子：
--
通讯作者：
Thomas Church;J. Ellenberg;B. Farb
中科院分区：
文献类型：
--
作者：
Thomas Church;J. Ellenberg;B. Farb
本文介绍并发展了fi模块的相关理论。我们运用这一理论来获得新的定理:——n截然不同的配置空间的上同调下令点任意(面向连接,)歧管- r组n的对角coinvariant代数变量的上同调和重复环的模空间n-pointed曲线的空间矩阵多项式秩品种的n * n -子代数的上同调属n Torelli集团由H ^ 1等等。对称群S_n作用于每一个向量空间。在大多数情况下，我们几乎不知道这些表征的特征，甚至不知道它们的维度。我们证明了在每个固定度中，当n足够大时，特征是由与n无关的循环计数函数中的多项式给出的。特别是，维度最终是n中的多项式。在这个框架中，s_n -表示序列的表示稳定性（在Church-Farb意义上）被转换为单个fi -模块的有限生成性质。
In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tautological ring of the moduli space of n-pointed curves - the space of polynomials on rank varieties of n x n matrices - the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module.