Structures in Representation Stability

Steven V. Sam

doi:10.1090/noti2011

Stability in the high-dimensional cohomology of congruence subgroups

Compositio Mathematica ◽

10.1112/s0010437x20007046 ◽

2020 ◽

Vol 156 (4) ◽

pp. 822-861

Author(s):

Jeremy Miller ◽

Rohit Nagpal ◽

Peter Patzt

Keyword(s):

Congruence Subgroup ◽

Stability Result ◽

High Dimensional ◽

Vanishing Theorem ◽

Congruence Subgroups ◽

Codimension One ◽

Representation Stability ◽

Steinberg Module ◽

Finiteness Properties ◽

Special Case

We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.

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Representation Stability of Power Sets and Square Free Polynomials

Canadian Journal of Mathematics ◽

10.4153/cjm-2014-029-2 ◽

2015 ◽

Vol 67 (5) ◽

pp. 1024-1045

Author(s):

Samia Ashraf ◽

Haniya Azam ◽

Barbu Berceanu

Keyword(s):

Symmetric Group ◽

Braid Group ◽

Point Of View ◽

Pure Braid Group ◽

Representation Stability ◽

Stability Point ◽

Power Set ◽

The Stability

AbstractThe symmetric group 𝓢n acts on the power set 𝓟(n) and also on the set of square free polynomials in n variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

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Homological Stability for Spaces of Commuting Elements in Lie Groups

International Mathematics Research Notices ◽

10.1093/imrn/rnaa094 ◽

2020 ◽

Cited By ~ 1

Author(s):

Daniel A Ramras ◽

Mentor Stafa

Keyword(s):

Lie Groups ◽

Lie Group ◽

Stable Range ◽

Rational Homology ◽

Infinite Dimensional ◽

Fixed Group ◽

Character Varieties ◽

Representation Stability ◽

Equivariant Homology ◽

Homological Stability

Abstract In this paper, we study homological stability for spaces $\textrm{Hom}({{\mathbb{Z}}}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $n\geqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, $\textrm{Comm}(G)$ and $B_{\textrm{com}} G$, introduced by Cohen–Stafa and Adem–Cohen–Torres-Giese, respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability—in particular, the theory of $\textrm{FI}_W$-modules developed by Church–Ellenberg–Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.

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