Stability in the homology of unipotent groups
Stability in the homology of unipotent groups
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DOI：
10.2140/ant.2020.14.119
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发表时间：
2017-11
影响因子：
1.3
通讯作者：
Andrew Putman;Steven V. Sam;A. Snowden
中科院分区：
文献类型：
--
作者：
Andrew Putman;Steven V. Sam;A. Snowden
设$R$是一个加法群是N-生成的环（不一定是交换的），$U_n（R）\子集GL_n（R）$是$R$上的上三角幂单矩阵群。从表示稳定性的角度研究了U_n（R）的同调群随n的变化规律。我们的主要定理断言，如果对于每个$n$，我们在环$\mathbf{k}$上有$U_n（R）$的表示$M_n$，它们是适当相容的，并且满足适当的有限性假设，那么规则$[n] \mapsto \widetilde{H}_i（U_n（R），M_n）$定义了一个n-生成的OI-模。因此，如果$\mathbf{k}$是一个域，那么$dim \widetilde{H}_i（U_n（R），\mathbf{k}）$最终等于$n$中的一个多项式。我们还证明了类似的结果为Iwahori子群$GL_n（\mathcal{O}）$的数环$\mathcal{O}$。
Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n(R) \subset GL_n(R)$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of $U_n(R)$ vary with $n$ from the point of view of representation stability. Our main theorem asserts that if for each $n$ we have representations $M_n$ of $U_n(R)$ over a ring $\mathbf{k}$ that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule $[n] \mapsto \widetilde{H}_i(U_n(R),M_n)$ defines a finitely generated OI-module. As a consequence, if $\mathbf{k}$ is a field then $dim \widetilde{H}_i(U_n(R),\mathbf{k})$ is eventually equal to a polynomial in $n$. We also prove similar results for the Iwahori subgroups of $GL_n(\mathcal{O})$ for number rings $\mathcal{O}$.