# Seminar page: Computations in Representation Theory

Description of the seminar: This seminar revolves around topics which are related to representation theory. It is an unusual seminar since emphasis is not on proofs but rather on computations. For example, speakers may:

(1) Present and interpret special cases of theorems

(2) Compute something interesting about some specific objects

(3) Make predictions about some specific objects

(4) Produce concrete formulae

(5) Geometrically interpret specific occurances

The above list are just examples and are not meant to exhaust options or limit the speaker in any way. While talks demonstrating computations relevant to original work are more than welcome, originality is not a requirement for a talk: For example, speakers may present special cases of well known results, or perform computations with well known results.

Mood of the Seminar: The seminar will be extremely informal, and we strongly encourage a multitude of questions and comments from attendees. Also, we do not mind at all if the speaker goes overtime, and thus excessive questions and comments will not limit the speaker in any way. Perhaps unlike other seminars, we welcome the speaker and attendees to chase tangential topics during the talk.

Time and Place: The Seminar is hosted every thursday from 5:30-6:30 pm.

Organizers: Yakov Kononov, Mrudul Thatte, Cailan Li and myself (Maithreya Sitaraman).

List of talks, Fall 2019:

Date Speaker Title Abstract
Sept 26 Yakov Kononov Some conjectures about knot polynomials
Oct 3 Maithreya Sitaraman New formulae for character columns of symmetric groups and their wreath products If I told you that we could compute character columns via nothing more than matrix multiplication, you probably would not believe me. However, in this talk I will show that this is possible!
Oct 10 Carl Lian Enumerating Pencils with Moving Ramification on Curves We consider the following problem: given an elliptic curve (E,p_1), how many degree d maps E -> P^1 are there ramifying to order d_1 at p_1 and orders d_2, d_3, d_4 at unspecified points p_2, p_3, p_4 of E? A geometric approach to this problem yields a formula that is strikingly simple, but that naturally includes contributions, with non-trivial multiplicities (products of Littlewood-Richardson coefficients), from linear systems with base-points. A somewhat mysterious inclusion-exclusion procedure then allows one to extract the answer to the original problem, and a surprising phenomenon arises: the answers are invariant under a certain involution. In other words, we get a "duality" for which I have no geometric explanation. I will explain this story, detailing some of the computations involved, and invite the audience to propose a representation-theoretic interpretation. If time permits, I will try to say something about generalizations to higher genus, via through the theory of limit linear series.
Oct 17 Fedor Kalinovich Popov (Physicist from Princeton) We study the fermionic quantum mechanical systems, where the fundamental degrees of freedom carries three indices and respect O(N)^3 symmetry. Such theories are integrable in the large N limit giving eventually an effective CFT in the IR. I will consider the case when one of the index is equal to 1 or 2 and show how from this one can derive Cauchy indentites for Schur polynomials and its generalizations for orthogonal Schur polynomials.
Oct 24 Cailan Li Representations of U_q(sl_2) when q is a root of unity We first classify all finite dimensional irreducible representations of U_q(sl_2) when q is a root of unity. We will then say a few words about what happens for general U_q(g) where g is a semisimple lie algebra, namely that the irreducible representations form a ramified covering of the corresponding Poisson dual group and the amazing connection to the modular representation theory of g for the corresponding restricted quantum group.
Oct 31 Konstantin Aleshkin Frobenius manifolds from singularity theory Mirror symmetry (in the enumerative meaning) can be formulated as "A-model of X = B-model of Y". One of the ways to define each of them is Frobenius manifolds which turn out to be quantum cohomology or generalization for the A-side and unfolding of singularity for the B-side. "Frobenius manifolds" on the spaces of unfoldings essentially appeared in the works of Kyoji Saito a decade before invention of the notion of Frobenius manifold. In the talk I will explain how this structure appear and present some examples of computations of these structures.
Nov 7 Joint seminar with conformal field thoery
Nov 14 Mrudul Thatte sl_n spider and the evaluation of MOY graphs We study the generators and relations of sl_n webs and obtain a diagrammatic presentation of the category of U_q(sl_n)-modules generated by the tensor products of fundamental representations. We use the relations in the sl_n spider to obtain a new definition of the evaluation of MOY graphs in terms of colorings.
Nov 21 Cancelled (Sam is sick)
Dec 5 Robin Zhang Gelfand pairs and their applications in arbitrary characteristic The theory of Gelfand pairs provides criteria for proving multiplicity one theorems in the study of branching problems. Outside of branching problems, Gelfand pairs have useful applications in the study of integrable systems, automorphic forms, and special values of L-functions. We present the classical theory of Gelfand pairs for complex representations and its application to some examples. We also introduce some extensions of the theory of Gelfand pairs to representations over algebraically closed fields of positive characteristic.
Dec 12 Sam DeHority Root multiplicities in Lorentzian Kac Moody Algebras and Siegel Modular Forms This talk will present two approaches to understanding imaginary root multiplicities in Kac-Moody algebras. The first is computational and quite general, while the second is by interpreting the Weyl-Kac character formula as an infinite product formula for some modular form, which works in some special and interesting cases.