Quantum Groups, Representation Theory, Superalgebras,
and Tensor Categories : an on-line conference
August 20 – 23, 2020
Conference organizers:
Hadi Salmasian
,
Siddhartha Sahi
.
Schedule of Talks
Time Zone: Eastern Standard Time (EST)
Thursday August 20
9:45–10:00
Opening
10:00–10:30
Stefan Kolb
Bivariate continuous q-Hermite polynomials and deformed quantum Serre relations.
Abstract
: In this talk I will explain how quantum symmetric pairs naturally give rise to a new family of bivariate continuous q-Hermite polynomials. The main tool is a star-product method which interprets the coideal subalgebras in the theory of quantum symmetric pairs as deformations of partial quantum parabolic subalgebras. It turns out that the defining relations for quantum symmetric pairs can be expressed in terms of continuous q-Hermite polynomials. The talk is based on joint work with Riley Casper and Milen Yakimov.
Slides
Video
10:30–10:50
Discussion break
10:50–11:20
Ben Webster
Tensor products and categorification
Abstract
: One key tool in understanding categories of representations of Lie (super)algebras and quantum groups is how the fun tour of tensor product with finite dimensional representations behaves. I’ll first explain how my work as well as that of many others has led to a good understanding of this in the type A case, and then say a few words about how we might generalize to BCD types.
Slides
Video
11:20–11:50
Discussion break
11:50–12:20
Vidya Venkateswaran
Quasi-polynomial representations of double affine Hecke algebras
Abstract
: In the 1990's, Cherednik introduced a Y-induced, cyclic representation of the double affine Hecke algebra on the space of polynomials, the so-called basic representation. In addition to its importance in the representation theory of DAHA, this representation plays an integral role in the theory of Macdonald polynomials. In this talk, we present a generalization of this picture. We study a class of $Y$-induced cyclic representations of DAHA, and show that they admit explicit realizations on the space of quasi-polynomials. We establish several properties about these representations, which parallel the basic representation, and we define a new family of quasi-polynomials which generalize Macdonald polynomials. We will also discuss some connections to recent work on Weyl group multiple Dirichlet series and metaplectic Whittaker functions. This is joint work with Siddhartha Sahi and Jasper Stokman.
Slides
Video
12:20–12:40
Discussion break
12:40–13:10
Vera Serganova
Representation of super Yangians of type Q.
Abstract
: The talk concerns classification of finite-dimensional irreducible representations of the Yangians associated with the Lie superalgebras Q(n), introduced by Nazarov. We present a complete classification for the case n=1 and some initial steps for solving the problem for n>1. (joint with E. Poletaeva)
Slides
Video
13:10–13:30
Discussion break
Friday August 21
10:00–10:30
Shifra Reif
Denominator identities for the periplectic Lie superalgebra p(n).
Abstract
: We will present the denominator identities for the periplectic Lie superalgebras and discuss their relations to representations of p(n) and gl(n). Joint work with Crystal Hoyt and Mee Seong Im.
Slides
Video
10:30–10:50
Discussion break
10:50–11:20
Jasper Stokman
N-point spherical functions
Abstract
: I will apply ideas from boundary Wess-Zumino-Witten conformal field theory to harmonic analysis on split real semisimple Lie groups. It leads to the introduction of N-point spherical functions as the appropriate analogues of N-point correlation functions for chiral vertex operators. I will show that N-point spherical functions solve a consistent system of first order differential equations. Various other properties of N-point spherical functions will be highlighted. This is joint work with N. Reshetikhin.
Slides
Video
11:20–11:40
Discussion break
11:40–12:10
Henrik Gustafsson
Whittaker functions and Yang-Baxter equations
Abstract
: We will discuss connections between the quantum group U
_{q}
(gl(r|n)) and Iwahori Whittaker functions on the metaplectic n-cover of GL_r(F) where F is a non-archimedean field. In particular, using a lattice model description we will illustrate how Yang-Baxter equations for the above quantum group recover the recursion relations for these Whittaker functions described by metaplectic Demazure operators. Based on joint work with Ben Brubaker, Valentin Buciumas and Daniel Bump.
Slides
Video
12:10–12:30
Discussion break
Saturday August 22
10:00–10:30
Alistair Savage
Categorical comultiplication
Abstract
: We will describe an analogue of comultiplication for certain monoidal categories. We will start with simple examples categorifying the standard comultiplication for symmetric functions, before treating Heisenberg categories. We will then explain how categorical comultiplication is a very useful tool for proving basis theorems for monoidal categories.
Slides
Video
10:30–10:50
Discussion break
10:50–11:20
Dimitar Grantcharov
Quantized enveloping superalgebra of type P.
Abstract
: We will introduce a new quantized enveloping superalgebra attached to the periplectic Lie superalgebra p(n). This quantized enveloping superalgebra is a quantization of a Lie bisuperalgebra structure on p(n). Furthermore, we will introduce the periplectic q-Brauer algebra and see that it admits natural centralizer properties. This is joint work with S. Ahmed and N. Guay.
Slides
Video
11:20–11:40
Discussion break
11:40–12:10
Alexander Sherman
Ghost Distributions on Supersymmetric Spaces.
Abstract
: We introduce ghost distributions on a supersymmetric space. They generalize the ghost centre of the enveloping algebra of a Lie superalgebra, as defined by Maria Gorelik, to supersymmetric pairs. Ghost distributions are invariant under a certain Lie superalgebra, and can be identified, as a vector space, with the invariant differential operators of the underlying symmetric space. We discuss what is known about the image of these distributions under the Harish-Chandra homomorphism, and what representation-theoretic implications it has. Finally, we mention when and how one can lift these distributions to (differential) operators on the supersymmetric space.
Slides
Video
12:10–12:30
Discussion break
Sunday August 23
10:00–10:30
Huanchen Bao
Flag manifolds over semifields.
Abstract
: The study of totally positive matrices, i.e., matrices with positive minors, dates back to 1930s. The theory was generalised by Lustig to arbitrary reductive groups using canonical bases, and has significant impacts on the theory of cluster algebras, totally positive flag manifolds, etc. In this talk, we review basics of total positivity and explain its generalization to general semifields. This is based on joint work with Xuhua He.
Slides
Video
10:30–10:50
Discussion break
10:50–11:20
Johannes Flake
Interpolation tensor categories, partition quantum groups, and monoidal centers.
Abstract
: Deligne showed that from interpolating families of representation categories, one obtains interesting examples of (not necessarily abelian) tensor categories. We will review the construction and its properties for the family of all symmetric groups. I will then explain some joint work with Laura Maaßen on certain subcategories related to (partition) quantum groups, and some joint work with Robert Laugwitz on the monoidal centers of these interpolation categories.
Slides
Video
11:20–11:40
Discussion break
11:40–12:10
Bogdan Ion
Stable DAHA’s and the double Dyck path algebra.
Abstract
: The double Dyck path algebra (ddpa) is the algebraic structure that governs the phenomena behind the shuffle and rational shuffle conjectures. I was introduced by Carlsson and Mellit as the key character in their proof of the shuffle conjecture and later Mellit used it to give a proof of the rational shuffle conjecture. While the structure emerged from their considerations and computational experiments while attacking the conjecture, it bears some resemblance to the structure of a double affine Hecke algebra (daha) of type A. Carlsson and Mellit mentioned the clarification of the precise relationship as an open problem. I will explain how the entire structure emerges naturally and canonically from a stable limit of the family of $GL_n$ daha’s. From this perspective a new commutative family of operators emerges. Their spectral properties are still to be explored. This is joint work with Dongyu Wu.
Slides
Video
12:10–12:30
Discussion break
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Hadi Salmasian
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