Tag - Representation theory

Pramod Achar: Cleanness in Springer theory

30th May, 2024

In Lusztig's papers from 1985-1986 that invented the theory of character sheaves, he proved (in nearly all cases) a remarkable property of cuspidal perverse Q-sheaves on the nilpotent variety: they are 'clean', meaning that their stalks vanish outside a single orbit. This property is crucial to making character sheaves computable by an algorithm, and it is a precursor of various 'block decompositions' of the derived category studied by various authors (Gunningham, Rider, Russell, and others) later. About 10 years ago, Mautner conjectured that these perverse sheaves remain clean after reduction modulo p (with some exceptions for small p). In this talk, I will discuss the history and context of the cleanness phenomenon, along with recent progress on Mautner’s conjecture.

Karin Erdmann: The Hemmer-Nakano Theorem and relative dominant dimension

30th May, 2024

Let ℋq(d) be the Iwahori-Hecke algebra of the symmetric group where q is a primitive ℓ-th root of unity, and let A = Sq(n,d) be the q-Schur algebra. Hemmer and Nakano proved amongst others that for ℓ ≥ 4, the Schur functor gives an equivalence between the category of A-modules with Weyl filtration, and the category of ℋq(d)-modules with dual Specht filtration, and that certain extension groups get identified. This has been a surprise and has inspired further research. In this talk we discuss some extensions of this result.

Dave Benson: The nucleus and the singularity category of cochains on the classifying space

30th May, 2024

The definition of the nucleus was originally formulated in joint work with Carlson and Robinson, to capture the supports of modules with no cohomology. This definition works in various contexts such as finite groups, restricted Lie algebras, and more generally, suitable triangulated categories of modules. In the finite group context it has a characterization in terms of subgroups whose centralizer is not p-nilpotent. In the restricted Lie algebra context, it is described in terms of the Richardson orbit, at least for large primes. Recent work with Greenlees has highlighted a connection with the singularity category of the cochains on the classifying space, in the group theoretic context. My plan is to give an introduction to these ideas.

Vera Serganova: Sylow theorems and support varieties for supergroups

30th May, 2024

We discuss support variety theory for quasireductive algebraic supergroups, i.e. supergroups with reductive even part over complex numbers. The corresponding categories of representations are Frobenius and share many properties of representations of finite groups in positive characteristic. It is desirable to describe Balmer spectrum of related triangulated symmetric monoidal categories. Our approach involves so called homological odd elements and certain tensor functors associated to them. On the way we encounter analogues of p-groups and Sylow subgroups for supergroups. We prove projectivity detection for our support theory and present other related results. We also explore connections with homological support theory developed by B. Boe, J. Kujawa and D. Nakano.

Shrawan Kumar: A new diagrammatic categorical setting for Schur dualities

29th May, 2024

Let Γ be a finite group acting on a simple Lie algebra 𝔤 and acting on an s-pointed projective curve (Σ, p = {p1, . . . ,ps}) faithfully (for s ≥ 1). Also, let an integrable highest weight module Hc(λi) of an appropriate twisted affine Lie algebra determined by the ramification at pi with a fixed central charge c is attached to each pi. We prove that the space of twisted conformal blocks attached to this data is isomorphic to the space associated to a quotient group of Γ acting on 𝔤 by diagram automorphisms and acting on a quotient of Σ. Under some mild conditions on ramification types, we prove that calculating the dimension of twisted conformal blocks can be reduced to the situation when Γ acts on 𝔤 by diagram automorphisms and covers of ℙ1 with 3 marked points. Assuming a twisted analogue of Teleman's vanishing theorem of Lie algebra homology, we derive an analogue of the Kac-Walton formula and the Verlinde formula for general Γ-curves (with mild restrictions on ramification types). In particular, if the Lie algebra 𝔤 is not of type D4, there are no restrictions on ramification types.

Chun-Ju Lai: Quantum Wreath Products

29th May, 2024

A quantum wreath product is the algebra produced from a given (not necessarily commutative) algebra B, a positive integer d, and a choice of certain coefficients in B ⊗ B. Important examples include variants of the Hecke algebras, such as (1) affine Hecke algebras and their degenerate version, (2) Wan-Wang’s wreath Hecke algebras, (3) Kleshchev-Muth’s affinization algebras, (4) Rosso-Savage’s (affine) Frobenius Hecke algebras, (5) endomorphism algebras arising from Elias’s Hecke-type categories, (6) Mathas-Stroppel’s Rees affine Frobenius Hecke algebras, and (7) Hu algebra, which quantizes the wreath product Sm ≀ S2 between the symmetric groups. Our goal is to develop a uniform approach to the structure and representation theory in order to encompass known results which were proved in a case by case manner. In this talk, I’ll focus on the Schur-Weyl duality and the Clifford theory. Our theory is motivated by (and has application to) the Ginzburg-Guay-Opdam-Rouquier problem on quasi-hereditary covers of Hecke algebras for complex reflection groups.

Jon Brundan: Isomeric Heisenberg and Kac–Moody categorification

29th May, 2024

The isomeric Heisenberg category acts naturally on a number of abelian categories appearing in the representation theory of the isomeric supergroup Q(n), and also on representations of Sergeev’s algebra which is related to the double covers of symmetric groups. I will explain an efficient way to convert an action of the isomeric Heisenberg category on these and other abelian categories into an action of a corresponding super Kac–Moody 2-category. To properly understand the odd simple root indexed by the element zero of the ground field requires the theory of odd symmetric functions developed by Ellis, Khovanov and Lauda, the quiver Hecke superalgebras of Kang, Kashiwara and Tsuchioka, and the covering quantum groups defined and studied by Clark and Wang.

Weiqiang Wang: A new diagrammatic categorical setting for Schur dualities

29th May, 2024

The classical Schur duality is a simple yet powerful concept which relates the representations of the symmetric group and general linear Lie algebra, as well as combinatorics of symmetric functions. This admits a quantum deformation to a duality between a quantum group and Hecke algebra of type A. In this talk, we will describe several new simple diagrammatic (monoidal/quotient) categories, where old and new algebras behind (affine/cyclotomic) Schur duality emerge naturally. Our construction has new combinatorial implications on symmetric functions and RSK correspondence.

Christopher Drupieski: The Lie superalgebra of transpositions

28th May, 2024

In this talk I will report on work, joint with Jonathan Kujawa, to answer a series of questions originally posed by MathOverflow user WunderNatur in August 2022: Considering the group algebra ℂSn of the symmetric group as a superalgebra (by considering the even permutations in Sn to be of even superdegree and the odd permutations in Sn to be of odd superdegree), and then in turn considering ℂSn as a Lie superalgebra via the super commutator, what is the structure of ℂSn as a Lie superalgebra, and what is the structure of the Lie sub-superalgebra of ℂSn generated by the transpositions? The non-super versions of these questions were previously answered by Ivan Marin, with very different results. Time permitting, some thoughts on analogues of these questions for Weyl groups of types B/C and D may also be discussed.

Eric Friedlander: Infinite-dimensional modules for linear algebraic groups

28th May, 2024

We investigate infinite-dimensional modules for a linear algebraic group 𝔾 over a field of positive characteristic p. For any subcoalgebra C ⊂ 𝒪(𝔾) of the coordinate algebra of 𝔾, we consider the abelian subcategory CoMod(C) ⊂ Mod(𝔾) and the left exact functor (−)C : Mod(𝔾) → CoMod(C) that is right adjoint to the inclusion functor. The class of cofinite 𝔾-modules is introduced using finite-dimensional subcoalgebras of 𝒪(𝔾). We categorify a construction of Hardesty-Nakano-Sobaje, thereby supplementing cofinite type in providing invariants for proper mock injective 𝔾-modules.