The UC Berkeley Representation theory and tensor categories seminar |
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| 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 graphsAbstract: 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 superalgebraAbstract: 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 CategoriesAbstract: 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 sungularitiesAbstract: 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 theoryAbstract: 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 CategoryAbstract: 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 |
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| April 22 | Agustina Czenky, USC |
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| April 29 | Pablo S. Ocal, OIST |
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