Categorical Representation Theory Seminar (Spring 2024)

This is a continuation of last semester's learning/research seminar on representation theory, often with an eye towards categorification. Talks will likely be disconnected talks from across representation theory and categorifcation, reflective of what the speaker is currently interested in working on/exploring; in particular this seminar is not cumulative and you are welcome to attend whatever talk you find interesting. There may be talks relating to affine Hecke algebras, tensor categories, Brauer categories, Soergel bimodules, etc..

Talks will roughly be 45 minutes followed by a 5 minute break, followed by 45 more minutes. Please email Cailan at ccl2166@columbia.edu if you'd like to give a talk.

Schedule

01/24
Speaker: Cailan Li
Title: Kazhdan-Lusztig Theory and Highest-Weight Categories
Abstract: We define abstract Kazhdan-Lusztig theories for highest weight categories following [CPS]. We then give two applications, the first allows us to calculate dimensions of Ext groups between any two simple modules in our category (in particular for Category O). The second gives us a criteria for complete reducibility via "parity" considerations and was one of the motivations for the notion of "parity sheaves." (cf. this and this and this)
01/31
Speaker: Fan Zhou
Title: Representation Theory of Quivers
Abstract: We discuss the representation theory of quivers (with relations), pointing out the constructions of the simples and the projectives/injectives. Every basic finite-dimensional algebra over an algebraically closed field is isomorphic to the path algebra of a quiver, and we say some words on how to find this quiver.
02/07
Speaker: Alvaro Martinez
Title: SL_2-Plethysms, Old and New
Abstract: Much of the history of representation theory and algebraic combinatorics has been concerned with the study of plethysms, which are compositions of Schur functors. Finding formulas for these has been one of the major open problems in mathematics. When evaluated on a representation of $SL_2$, the situation becomes more tractable, yet it is still very rich, especially over arbitrary fields. We will give an overview of the classical results, which go back to the mid-19th century, up to some closely related recent developments.
02/14
Speaker: Cailan Li
Title: Clifford Theory for Crossed Product Algebras
Abstract: Given an action of a group $G$ on an algebra $R$, one can form the crossed product algebra $R\rtimes G$. In preparation for future talks on representations of affine Hecke algebras, we will go over Clifford Theory for Crossed Products algebras, which allows us to construct the simples for $R\rtimes G$ or $R^G$ from the simples of $R$ and $G$. Time permitting, we might talk a bit on classical Clifford Theory for finite groups.
02/21
Speaker: Pavel Turek
Title: Schur functors and plethysms in the stable module category of SL_2(F_p) in characteristic p
Abstract: Schur functors are endofunctors in categories of modules of an algebra. They can be used to construct all polynomial representations of GL(V) in characteristic 0 and to recover Schur polynomials by considering formal characters. A still unanswered question asks to describe compositions of Schur functors, the so-called modular plethysms. We consider this question for the natural two-dimensional module of SL_2(F_p) in characteristic p when the modular plethysms behave particularly nicely and classify ‘small’ modular plethysms which are projective, respectively, projective after forgetting one indecomposable summand. Using endotrivial modules we then find the stable representation ring of SL_2(F_p) and describe how Schur functors act on indecomposable modules of SL_2(F_p).
02/28
Speaker: Chris Bowman
Title: Combinatorics and Stabilities of Plethysms
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03/06
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03/13
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03/20
Speaker: Jonathan Brundan
Title: TBD
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03/27
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04/03
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04/10
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04/17
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