## Representation Theory and Applications

##### Session Organizers

Robert M. Guralnick, University of Southern California

Arun Ram, University of Melbourne

Pham H. Tiep, Rutgers University

### 5 December, 2022

##### Abstract

In 1955, Richard Brauer made what was arguably the first “local-global” conjecture in character theory. I’ll discuss the conjecture, often known as Brauer’s Height Zero Conjecture, and its recent proof due to joint work with G. Malle, G. Navarro, and P.H. Tiep.

##### Abstract

Complex irreducible characters of finite groups have two main invariants that are used to measure their rationality: their character fields and their Schur indices. We will discuss recent work aiming to explicitly compute these invariants for certain families of finite reductive groups, in particular the symplectic, special orthogonal, and unitary groups.

##### Abstract

This is based on joint work with Masoud Kamgarpour and GyeongHyeon Nam. Character varieties are geometric objects associated to maps from a finitely generated group to a reductive group. Seminal work by Hausel and Rodriguez-Villegas studied topological properties of character varieties by counting points over finite fields. They considered twisted GL(n) character varieties of the fundamental group of certain surfaces. In this talk, we will discuss character varieties associated to an orientable surface with punctures, and a reductive group G with connected centre. We give a sufficient condition for their smoothness. Assuming regular semisimple, and regular unipotent monodromy, we give an explicit, polynomial point count. This allows us to determine the Euler characteristic and the number of connected components.

##### Abstract

This talk will be about the interface of representation theory and machine learning. In machine learning, one sometimes wants to learn quantities which are invariant or equivariant with respect to a group. For example, the decision as to whether there is a tiger nearby should not depend on the precise position of your head and thus this decision should be rotation invariant. Another example: quantities that appear in the analysis of point clouds often do not depend on the labelling of the points, and are therefore invariant under a large symmetric group. I will explain how to build networks which are equivariant with respect to a group action. What ensues is a fascinating interplay between group theory, representation theory and deep learning. Examples based on translations or rotations recover familiar convolutional neural nets, however the theory gives a blueprint for learning in the presence of complicated symmetry. These architectures appear very useful to mathematicians, but I am not aware of any major applications in mathematics as yet. Most of this talk will be a review of ideas and techniques well-known in to the geometric deep learning community. New material is joint work with Joel Gibson (Sydney) and Sebastien Racaniere (DeepMind).

##### Abstract

I will discuss the irrationality at a prime $p$ of the values of irreducible characters of a finite group. Among $p$’-degree characters, this irrationality behaves very nicely due to its conection with Navarro’s refinement of the celebrated McKay conjecture. I will present some recent work on the continuity of $p$-irrationality of $p$’-degree characters, bounding almost $p$-rational characters, and bounding odd-degree characters. Some results are joint with G. Malle and A. Maroti.

##### Abstract

The Springer theory relates nilpotent orbits in the Lie algebra of a reductive algebraic group to irreducible representations of Coxeter groups. We discuss a Springer theory for graded Lie algebras and the character sheaves arising in this setting, concentrating on the construction of cuspidal character sheaves. Irreducible representations of Hecke algebras of complex reflection groups at roots of unity enter the description of character sheaves. We will explain the connection between our work and the recent work of Lusztig and Yun, where irreducible representations of trigonometric double affine Hecke algebras appear in the picture. This is based on joint work with Kari Vilonen and partly with Misha Grinberg.

### 6 December, 2022

##### Abstract

Langlands’ reciprocity conjecture parameterizes smooth irreducible representations of a reductive group $G$ over a local field, or automorphic representations of a reductive group $G$ over a global field, in terms of the $L$-group of $G$, which is a split extension of the dual group of $G$ and the Galois group. Langlands’ functoriality conjecture predicts relationship between such representations given a homomorphism between the $L$-groups. In practice, one is often confronted with extensions of the dual group by the $L$-group that are not split. We will explore this phenomenon based on the following construction. To a connected reductive group $G$ over a local field $F$ we define a compact topological group $\tilde\pi_1(G)$ and an extension $G(F)_\infty$ of $G(F)$ by $\tilde\pi_1(G)$. From any character $x$ of $\tilde\pi_1(G)$ of order $n$ we obtain an $n$-fold cover $G(F)_x$ of the topological group $G(F)$. We also define an $L$-group for $G(F)_x$, which is a usually non-split extension of the Galois group by the dual group of $G$, and deduce (from the linear case) a refined local Langlands correspondence between genuine representations of $G(F)_x$ and $L$-parameters valued in this $L$-group. We will present one concrete application: a characterization of the local Langlands correspondence for semi-simple discrete $L$-parameters, that is uniform for all local fields.

##### Abstract

Tilting modules for a reductive algebraic group G in prime characteristic were formally introduced (in this setting) by Donkin in the early 1990’s. They have since come to play an increasingly central role in the study of G-modules, featuring prominently in modular character formulas and also shedding light on the “Humphreys-Verma Problem.”

In the early days, Donkin made a series of conjectures that predicted properties of the direct summands of tensor products involving the Steinberg module. Some of these conjectures remain open, while others have been resolved only in the last few years. In this talk I will discuss what we presently know and do not know, and why it is important. This is based on joint work with Chris Bendel, Dan Nakano, and Cornelius Pillen.

##### Abstract

The collection of parking functions under a natural Sn-action (which has Catalan-many orbits) has been a central object in Algebraic Combinatorics since the work of Haiman more than 30 years ago. One of the lines of research spawned around it was towards defining and studying analogous objects for real and complex reflection groups W; the main candidates are known as the W-non-nesting and W-non-crossing parking functions.

The W-non-nesting parking functions are relatively well understood; they form the so called algebraic W-parking-space which has a concrete interpretation as a quotient ring (of the ambient polynomial algebra over a system of homogeneous parameters that carry the reflection representation of W). The W-non-crossing parking functions on the other hand have defied unified explanations while simultaneously proving themselves central in the study of Coxeter and Artin groups (their geometric group theory and combinatorics) and in the representation theory of Dynkin quivers. One of the main open problems in the field since the early 2000’s has been to give a type-independent proof of the W-isomorphism between the algebraic and the non-crossing W-parking spaces. In this talk, I will present the first such proof, solving the more general Fuss version of the problem, that proceeds by comparing a collection of recursions that are shown to be satisfied by both objects. This relies on a variety of recent techniques we introduced, in particular a parabolic decomposition of the algebraic parking space and its relation with the spectrum of Laplacian matrices for reflection arrangements.

##### Abstract

The irreducible representations of the Symmetric group are a classical subject, seemingly well understood. Yet, the multiplicities in the irreducible decomposition of the tensor product of two irreducibles, the Kronecker coefficients, present an 80+ long standing open problem. The problem has seen its revival in the context of Geometric Complexity Theory for the separation of complexity classes, the algebraic analogues of P vs NP.

In this talk we will review some recent results on the asymptotics and complexity of Kronecker coefficients and the underlying characters of the Symmetric group. Based on a series of joint works with Christian Ikenmeyer and Igor Pak.

##### Abstract

A Kronecker coefficient $g(\lambda, \mu, \nu)$ is a non-negative integer that depends on three partitions $\lambda$, $\mu$, $\nu$ of a natural number $n$. It is the multiplicity of an irreducible representation $V^\nu$ of the symmetric group of degree n in the tensor (or Kronecker) product $V^\lambda \otimes V^\mu$ of two other irreducible representations of the same group.

The study of ways of computing Kronecker coefficients is an important topic on algebraic combinatorics. Several tools have been used to try to understand them, notably from representation theory, symmetric functions theory and Borel-Weil theory. These numbers generalize the well-known Littlewood-Richardson coefficients but are still very far to be fully grasped.

It is known that each Kronecker coefficient can be described as an alternating sum of numbers of integer points in convex polytopes. In this talk we present a new family of polytopes that permit efficient computations on Kronecker coefficients associated to partitions with few parts and provides insight in the behavior of Murnaghan stability.

##### Abstract

The dual Specht module of the symmetric group algebra over $\mathbb{Q}$ has two distinguished bases, namely the standard basis and Young’s seminormal basis. We study how the Young’s seminormal basis vectors are expressed in terms of the standard basis, as well as the denominators of the coefficients in these expressions. We obtain closed formula for some Young’s seminormal basis vectors, as well as partial results for the denominators in general.

This is a joint work with Ming Fang (Chinese Academy of Sciences) and Kay Jin Lim (Nanyang Technological University).