TAPIRS

I will introduce the notion of homological stability for algebras, and present work on homological stability for two families of diagram algebras: the Temperley-Lieb algebras and the Brauer algebras. I will sketch where the proof of homological stability for algebras (possibly) diverges from the usual framework of homological stability proofs, and touch on some interesting connections between diagram algebras and other fields. This talk will be accessible to a broad audience in topology and related fields. The results I will present are joint work with Hepworth for Temperley-Lieb algebras and Hepworth and Patzt for Brauer algebras. All relevant papers can be found on the arXiv.

It was discovered by Galatius–-Kupers–-Randal-Williams that the failure of homological stability itself can exhibit stability phenomena, dubbed secondary homological stability. In the setting of representation stability, Miller--Wilson showed that a similar phenomenon, secondary representation stability, occurs for the homology of (ordered) configuration spaces for non-compact surfaces. We will formulate and prove higher representation stability results for the cohomology of (generalized) configuration spaces of a scheme/topological space. These stability results have a slightly different flavor from the one proved by Miller--Wilson. The main tool we employ is the theory of factorization homology with coefficients in twisted commutative algebras (TCAs). The results themselves are formulated in the language of derived indecomposables in the sense of Galatius–-Kupers–-Randal-Williams. Both of these will be reviewed in the talk.

VIC(R) is an analog of the category FI with the role of symmetric groups S_n replaced by the general linear groups GL_n(R). As such, a VIC(R)-module can be thought of as a sequence of general linear group representations satisfying certain natural compatibility conditions. I will give an overview of the representation theoretic aspects of this theory for various commutative rings R and various classes of VIC-modules.

The Steinberg representation is an important representation of a reductive group like GL_n. It arose classically in work of Steinberg, who proved that for a finite field k, the Steinberg representation of GL_n(k) is a finite-dimensional irreducible representation. For infinite k, it is an infinite-dimensional representation of GL_n(k) that due to work of Borel-Serre plays a basic role in the cohomology of arithmetic groups. Recently this representation has attracted the attention of the representation stability community. In this talk, I will discuss recent joint work with Andrew Snowden in which we prove that for infinite fields k, the Steinberg representation of GL_n(k) is irreducible. Since this is an infinite-dimensional representation of an infinite group, the proof is by necessity very different from the classical proof of Steinberg for finite fields.

Given a module, M, over a combinatorial category, what can we say about its underlying sequence of group representations? I will talk about a strategy for answering this question, by using poset topology to associate chain complexes to M. I will concentrate on the case where the combinatorial category is FS^op, the category of finite sets and surjections, where this strategy lets us describe the character of M.