Tag - Groups of Lie type

Pavel Turek: On stable modular plethysms of the natural module of SL2(𝔽p) in characteristic p

24th January, 2023

To study polynomial representations of general and special linear groups in characteristic zero one can use formal characters to work with symmetric functions instead. The situation gets more complicated when working over a field k of non-zero characteristic. However, by describing the representation ring of kSL2(𝔽p) modulo projective modules appropriately we are able to use symmetric functions with a suitable specialisation to study a family of polynomial representations of kSL2(𝔽p) in the stable category. In this talk we describe how this introduction of symmetric functions works and how to compute various modular plethysms of the natural kSL2(𝔽p)-module in the stable category. As an application we classify which of these modular plethysms are projective and which are 'close' to being projective. If time permits, we describe how to generalise these classifications using a rule for exchanging Schur functors and tensoring with an endotrivial module.

Yifan Jing: Measure Growth in Compact Simple Lie Groups

8th November, 2022

The celebrated product theorem says if A is a generating subset of a finite simple group of Lie type G, then |AAA| ≫ min ( |A|1+c, |G| ). In this talk, I will show that a similar phenomenon appears in the continuous setting: If A is a subset of a compact simple Lie group G, then μ(AAA) > min ( (3+c)μ(A), 1 ), where μ is the normalized Haar measure on G. I will also talk about how to use this result to solve the Kemperman Inverse Problem, and discuss what will happen when G has high dimension or when G is non-compact.

Inna Entova-Aizenbud: Representation stability for GLn(𝔽q)

10th May, 2022

I will present some results from a work in progress joint with Thorsten Heidersdorf on the Deligne categories for the family of groups GLn(𝔽q), for non-negative integers n. The Deligne categories interpolate the tensor categories of complex representations of GLn(𝔽q), and have been previously constructed by F. Knop and E. Meir (for certain values of n). I will describe some properties of these categories as well as their relation to the category of algebraic representations of the infinite group GL∞(𝔽q).

Michael Larsen: Quotients of normal subsets in simple groups

7th October, 2021

Let G be a finite simple group and S a normal subset of G. If |G| is large enough in terms of |S|/|G|, can we deduce that every element of G can be expressed as x y-1 for x and y elements of S? Shalev, Tiep, and I have proven that this is true assuming G is an alternating group or a group of Lie type in bounded rank, but the question remains open for classical groups of high rank over small fields. I will say something about the methods of proof, which involve both character methods and geometric ideas and also say something about the more general question of covering G by ST where S and T are large normal subsets.

David Craven: Maximal subgroups of finite simple groups

20th August, 2021

In this talk we will discuss the structure of maximal subgroups of finite simple groups, particularly groups of Lie type. We will discuss subgroups of exceptional groups of Lie type, and a version of Ennola duality that exists for groups of Lie type, which relates untwisted and twisted groups of Lie type.

Tasho Kaletha: An explicit supercuspidal local Langlands correspondence

29th October, 2020

We will give an explicit construction and description of a supercuspidal local Langlands correspondence for any p-adic group G that splits over a tame extension, provided p does not divide the order of the Weyl group. This construction matches any discrete Langlands parameters with trivial monodromy to an L-packet consisting of supercuspidal representations, and describes the internal structure of these L-packets.

The construction has two parts. The depth-zero part involves generalizing to disconnected groups results of Lusztig on the decomposition of a non-singular Deligne-Lusztig induction. Higher multiplicities occur in this decomposition and are handled using work of Bonnafé-Dat-Rouquier. The positive-depth part involves functorial transfer from a twisted Levi subgroup, which is made possible by an improvement of Yu's construction of supercuspidal representations obtained in recent joint work with Fintzen and Spice, and consideration of Harish Chandra characters.

We will also discuss ongoing work towards related conjectures: Shahidi's generic L-packet conjecture, Hiraga-Ichino-Ikeda formal degree conjecture,  stability and endoscopic transfer.

Aner Shalev: Groups in Interaction

18th September, 2015

This video is of the London Mathematical Society and European Mathematical Society‘s Joint Mathematical Weekend in 2015.