The UC Berkeley Representation theory and tensor categories seminar
Spring 2025 - Tuesday 2:10pm - 3:30pm, Evans 939
Organizers: Vera Serganova, Ilia Nekrasov, and Alexandra Utiralova

If you would like to be added to the seminar mailing list, contact any of the organizers.

DATE SPEAKER TITLE (click to show abstract)
January 21 Nicolai Reshetikhin , UC Berkeley and BIMSA
Invariants of tangles with a flat connection in the complement I. Abstract: The construction of these invariants based on properties of quantum groups at roots of unity and it was proposed in a joint work with R. Kashaev. It is related to homotopy quantum field theory by V. Turaev. In this talk, I will recall the construction of these invariants, and I will explain why Poisson structures on the center of quantum groups at a root of unity that appear are natural from the geometry of flat connections in the complement to a tangle.
January 28 No seminar
February 4 Nicolai Reshetikhin , UC Berkeley and BIMSA
Invariants of tangles with a flat connection in the complement II.

Continuation of the previous talk.

Abstract: The construction of these invariants based on properties of quantum groups at roots of unity and it was proposed in a joint work with R. Kashaev. It is related to homotopy quantum field theory by V. Turaev. In this talk, I will recall the construction of these invariants, and I will explain why Poisson structures on the center of quantum groups at a root of unity that appear are natural from the geometry of flat connections in the complement to a tangle.

February 11 Christian Gaetz, UC Berkeley
SL(n) web bases from hourglass plabic graphs Abstract: The SL(3) web basis is a special diagrammatic basis for certain spaces of tensor invariants developed in the late 90’s by Kuperberg as a tool for computing quantum link invariants. Since then this basis has found connections and applications to cluster algebras, dimer models, quantum topology, and tableau combinatorics. A main open problem has remained: how to find a basis replicating the desirable properties of this basis for SL(4) and beyond? I will describe joint work with Oliver Pechenik, Stephan Pfannerer, Jessica Striker, and Josh Swanson in which we construct such a basis for SL(4). Modified versions of plabic graphs and the six-vertex model and new tableau combinatorics will appear along the way.
February 18 Maria Gorelik, Weizmann Institute and Vladimir Hinich, University of Haifa
On the centre of a universal enveloping superalgebra Abstract: In contrast to semisimple Lie algebras the centre of a universal enveloping superalgebra is not a Noetherian ring. A popular example is the ring of supersymmetric polynomials. In this talk we will discuss several properties of this centre. We use the technique similar to Beilinson-Bernstein localization, presenting this ring as the global sections of a structure sheaf on a ringed space which is not an algebraic variety.
February 25 Ilia Nekrasov, UC Berkeley
The dichotomy between Model Theory and Tensor Categories Abstract: First, I will briefly remind you of the construction of tensor categories from oligomorphic groups (developed by A. Snowden and N.Harman). And then I will explain how to leverage the dichotomy between model-theoretic notions and their categorical counterparts. In particular, I will explain
(a) how distal structures naturally arise in our (tensor-categorical) context and
(b) where to look for model-theoretic analogs of tensor functors.
Everyone is welcome! Logicians and model theorists are insistently invited.
March 4 Dmytro Matvieievskyi, Kavli IMPU
Spherical unitary dual via quantized symplectic sungularities Abstract: Let G be a complex reductive algebraic group. Describing the spherical unitary dual of G is an old and classical important problem in representation theory. In this talk I will explain some ideas of how to approach this question by quantizing symplectic singularities, namely nilpotent coadjoint orbit closures and their suitable generalizations. This is an ongoing project with Ivan Losev and Lucas Mason-Brown.
March 11 Peng Zhou, UC Berkeley
Cutting-and-gluing in categorified representation theory Abstract: Khovanov-Lauda and Rouquier initiated the higher representation program, by categorifying (the negative part) \(U_q(g)^-\) to a monoidal category U. In this talk, I will first give a symplectic geometric realization of this monoidal category (https://arxiv.org/abs/2406.04258), then explain how to take tensor product of two monoidal module categories by study a 'disk with three stops'. This is work in progress joint with Mina Aganagic, Elise LePage, Yixuan Li.
March 18 Vera Serganova, UC Berkeley
Sergeev duality and projective representations of symmetric groups. Abstract: Sergeev duality is a generalization of Schur-Weyl duality where the group GL(N) is replaced by the supergroup Q(N). The centralizer of Q(N) in the n-th tensor power of the standard representation is the Sergeev superalgebra A(n) which is Morita equivalent to the spin symmetric group algebra. The latter gives all irreducible projective representations of the symmetric group S(n) with central charge -1. Irreducible polynomial representations of Q(N) and irreducible projective representations of S(n) are enumerated by the strict partitions of n with at most N parts. Characters of polynomial irreducible Q(N) representations are given by the Hall-Littlewood polynomials for the special value of the parameter, they form a basis in the ring of supersymmetric polynomials of type Q. There is an analogue of Frobenius formula. Using Jucys-Murphy elements in A(n) we construct primitive idempotents and bases in terms of shifted standard tableaux. We also discuss the relation to the fusion procedure with the R-matrix of type Q Yangians introduced by Nazarov. The last part of the talk is based on a recent preprint with I. Kashuba and A. Molev.
March 25 Spring Recess
April 1 Peter McNamara, University of Melbourne
The Spin Brauer Category Abstract: The Brauer category is a tool that controls the representation theory of (special) orthogonal Lie groups and Lie algebras. A drawback is it doesn't see the spin representations. We introduce and study a spin version, the Spin Brauer category, which sees the entire representation theory of type B/D Lie algebras. This is joint work with Alistair Savage.
April 8 Adam Dhillon, UC Berkeley
Integrable highest weight representations of S(1,2;a) Abstract: Finite growth Kac-Moody Lie superalgebras can be viewed as a super analogue of finite-dimensional semisimple and affine Kac-Moody Lie algebras. Most of these finite growth Kac-Moody Lie superalgebras have a symmetrizable Cartan matrix and typical integrable representations that satisfy a super analogue of the Kac-Weyl character formula, and proofs use the existence of the Casimir operator. In this talk, I'll address the integrable highest weight representations of S(1,2;a), a class of non-symmetrizable Kac-Moody Lie superalgebras of finite growth, showing that the Weyl-Kac character formula is valid generically, but not in general.
April 15 Monica Vazirani, UC Davis
Using DAHA to understand quantum D-modules in type A Abstract: We use Jordan's Schur-Weyl functor to understand quantum D-modules via the double affine Hecke algebra H in type A. In fact, we prove an equivalence of categories between strongly equivariant quantum D-modules and He-modules, where e is the (finite) sign idempotent. A key tool is understanding the quantum analogue of Hotta-Kashiwara D-modules, of which a special case is the ``quantum Springer sheaf." Applications of these results include understanding the skein algebra of the 2-torus in type A for generic parameters.

This is joint work with Sam Gunningham and David Jordan.

April 22 Pablo S. Ocal, OIST
Warped tensor products and noncommutative 2d topological quantum field theories Abstract: In this talk I will present an attempt to define some noncommutative 2d topological quantum field theories using deformations of Frobenius algebras. First, we will overview the importance and uses of 2d topological quantum field theories, as well as their equivalence to commutative Frobenius algebras. Then, we will consider an attempt at our goal using the deformations of coalgebras given by cotwisted tensor products, characterize when these are Frobenius algebras, and explain the resulting deficiencies. As a second attempt, I will introduce the warped tensor product of Frobenius algebras and characterize when these are Frobenius algebras. We will use this characterization to construct families of bifunctors that yield symmetric monoidal structures on the category of Frobenius algebras, and justify why they deserve to be called noncommutative 2d topological quantum field theories.

This is based on work with Amrei Oswald, Rohan Das and Julia Plavnik.

April 29 Agustina Czenky, USC
Unoriented 2-dimensional TQFTs and the category \(\mathrm{Rep}(S_t)\). Let \(k\) be an algebraically closed field of characteristic zero. The category of oriented 2-dimensional cobordisms can be understood in purely algebraic terms via a description by generators and relations; moreover, it is possible to recover from it the Deligne category \(\mathrm{Rep}(S_t)\), which interpolates the category of finite-dimensional representations of the symmetric group \(S_n\) from \(n\) a positive integer to any parameter \(t\) in \(k\). We show an analogous story happens in the unoriented case: via its description by generators and relations, we recover the generalized Deligne category \(\mathrm{Rep}(S_t \wr \mathbb Z_2)\), which interpolates the category of finite-dimensional representations of the wreath product \(S_t \wr \mathbb Z_2\).
May 6 Albert Schwarz , UC Davis
Quantum theory and its generalizations from classical mechanics with a restricted set of observables Abstract: We consider classical mechanics, assuming that our devices can measure only some observables ('observable observables"). We show that with the appropriate choice of "observable observables," we obtain quantum theory, and for other choices we obtain more general theories that can be analyzed in the framework of the geometric approach to quantum theory suggested in my papers several years ago.
May 13 Ilya Dumanski, MIT
Noncommutative resolutions and canonical bases Abstract: Kazhdan and Lusztig identified the affine Hecke algebra with equivariant K-theory of Steinberg variety of the dual group. Bezrukavnikov categorified this identification, and in particular, gave a geometric description of the Kazhdan--Lusztig canonical basis. It is given by classes of simple perverse modules over the non-commutative resolution. We propose an analogous construction for a different situation: resolution of affine Schubert variety in type A. We construct the non-commutative resolution and describe the corresponding basis in terms of the quantum loop group action. We emphasize a relation to categorical Howe duality and K-theoretic Satake.

Based on a work in progress, joint with E. Bodish and V. Krylov.

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