Tag - Representation theory

Charlotte Chan: Deligne-Lusztig theory: examples and applications

20th May, 2024

Geometry and representation theory are intertwined in deep and foundational ways. One of the most important instances of this relationship was uncovered in the 1970s by Deligne and Lusztig: the representation theory of matrix groups over finite fields is encoded in the geometry of a natural 'partition' of flag varieties. Recent developments have revealed rich connections between Deligne-Lusztig varieties and geometry studied in number-theoretic contexts. In this lecture series, we give an example-based tour of these ideas, focusing on how to extract concrete information from theory.

Paolo Stellari: Comparing the homotopy categories of dg categories and of A∞-categories

9th May, 2024

In this talk, we show that the homotopy category of (small) dg categories and the homotopy category of A∞-categories are equivalent (even from a higher categorical viewpoint). We will discuss several issues related to the various notions of unity and provide several applications. The main ones are about the uniqueness of enhancements for triangulated categories and a full proof of a claim by Kontsevich and Keller concerning a description of the category of internal Homs for dg categories.

Eloísa Grifo: Searching for modules that are not virtually small

2nd May, 2024

Pollitz gave a characterization of complete intersection rings in terms of the triangulated structure of their derived category, akin to the Auslander-Buchsbaum-Serre characterization of regular rings. In this talk, we will explore how to bring this characterization back to the world of modules, and discuss the role of cohomological support varieties in solving this problem.

Julia Plavnik: Remarks on the tensor product property for support varieties for finite tensor categories

22nd April, 2024

For non-semisimple tensor categories satisfying some finiteness conditions, support varieties are meaningful geometric invariants of objects. Their theory began in the work of Quillen and Carlson on finite group representations. In more recent years, the theory of support varieties was generalized in many directions, including representations of finite-dimensional Hopf algebras and self-injective algebras, and objects in finite tensor categories and triangulated categories, among others.

In this talk, we will start by introducing the definition of support varieties for finite tensor categories and some of their basic properties. We will also present some conditions under which the tensor product property holds for support varieties, and we will present some applications to certain Hopf algebras. We will also discuss a construction of non-semisimple finite tensor categories with finitely generated cohomology for which the tensor product property does not hold for support varieties.

Eoghan McDowell: Spin representations of the symmetric group which reduce modulo 2 to Specht modules

16th April, 2024

When do two ordinary irreducible representations of a group have the same p-modular reduction? In this talk I will address this question for the double cover of the symmetric group, and more generally give a necessary and sufficient condition for a spin representation of the symmetric group to reduce modulo 2 to a multiple of a Specht module (in the sense of Brauer characters or in the Grothendieck group). I will explain some of the techniques used in the proof, including describing a function which swaps adjacent runners in an abacus display for the labelling partition of a character.

Nham Ngo: Restricted partitions and SL2-cohomology

1st April, 2024

In this talk, we first present our study on the number of partitions of a positive integer m into at most n parts in a given set A. We prove that such a number is bounded by the nth Fibonacci number F(n) for any m and some family of sets A including sets of powers of an integer. Then we use this result to estimate the cohomology space of the simple algebraic group SL2 with coefficients in Weyl modules. In particular, let k be an algebraically closed field of prime characteristic p and V(m) the Weyl SL2-module of highest weight m. We show that for p ≥ 5, dim Hn(SL2,V(m))≤ F(n+1) for all m,n ≥ 0.

Maxime Ramzi: Categorifying spectra and the theorem of the heart

28th March, 2024

The goal of this talk will be to present the results from my recent joint work with Vova Sosnilo and Christoph Winges, where we prove that every spectrum is the (non-connective) K-theory spectrum of a stable category. Our main application of this is the disproof of a conjecture by Antieau-Gepner-Heller about a non-connective version of the theorem of the heart in the non-noetherian setting; but I will also try to mention other perspectives on this result.

Jessica Fintzen: An introduction to representations of p-adic groups

25th March, 2024

An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool not just within representation theory, but also for the construction of an explicit and a categorical local Langlands correspondence, and has applications to the study of automorphic forms, for example. In my talk I will introduce p-adic groups and explain that the category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. I will then provide an overview of what we know about the structure of these Bernstein blocks. In particular, I will sketch how to use a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.

Naihuan Jing: Characters of GLn(q) and vertex operators

3rd March, 2024

Irreducible characters of the finite group GLn(q) were determined by Green in a remarkable paper that has influenced representation theory greatly. In this talk, I will discuss a vertex algebraic approach to construct and compute all complex irreducible characters of GLn(q). Green's theory is recovered and enhanced under the realization of the Grothendieck ring of representations R(G)=⨁​n≥0​​R(GLn(q)) as two isomorphic Fock spaces. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. This offers a simplification to identify the Fock space R(G) as the Hall algebra of symmetric functions. We will also discuss how to compute the characters in general.

Thomas Haines: Pavings of convolution fibres and applications

1st March, 2024

A convolution morphism is the geometric analogue of the convolution of functions in a Hecke algebra. The properties of fibres of convolution morphisms are used in a variety of ways in the geometric Langlands programme and in the study of Schubert varieties. I will explain a very general result about cellular pavings of fibres of convolution morphisms in the setting of partial affine flag varieties, as well as applications related to the very purity and parity vanishing of cohomology of Schubert varieties over finite fields, structure constants for parahoric Hecke algebras, and the (motivic) geometric Satake equivalence.