Tag - Representation theory

Mahir Can: Spherical Varieties and Combinatorics

29th September, 2020

Let G be a reductive complex algebraic group with a Borel subgroup B. A spherical G-variety is an irreducible normal G-variety X where B has an open orbit. If X is affine, or if it is projective but endowed with a G-linearized ample line bundle, then the group action criteria for the sphericality is in fact equivalent to the representation theoretic statement that a certain space of functions (related to X) is multiplicity-free as a G-module. In this talk, we will discuss the following question about a class of spherical varieties: if X is a Schubert variety for G, then when do we know that X is a spherical L-variety, where L is the stabilizer of X in G.

Eugene Gorsky: Link homology and Hilbert schemes

28th September, 2020

The category of Soergel bimodules categorifies the Hecke algebra. Khovanov and Rozansky used Soergel bimodules to define triply graded link homology (also known as Khovanov-Rozansky homology) which categorifies HOMFLY-PT link invariant. I will survey some results and conjectures relating Soergel bimodules and Khovanov-Rozansky homology with the Hilbert scheme of points on the plane. The talk is based on joint works with Matt Hogancamp, Andrei Negut, Jake Rasmussen and Paul Wedrich.

Alexander Premet: Modular representations of Lie algebras and Humphreys’s Conjecture

25th September, 2020

Let G be a connected reductive algebraic group defined over an algebraically closed field of positive characteristic p and suppose that the derived subgroup of G is simply connected, p is a good prime for the root system of G and the Lie algebra 𝔤=Lie(G) admits a non-degenerate Ad G-invariant symmetric bilinear form. If G is a simple algebraic group of type other than A, the above assumptions mean that p is a good prime for G, i.e. p ≥ 3 if G is of type B, C or D, p ≥ 5 if G is of type G2, F4, E6 or E7, and p ≥ 7 if G is of type E8. If all components of G have type A, B, C, D we set R=ℤ[1/2].  If G has a component of exceptional type but has no components of type E8 we set R=ℤ[1/6]. If G has a component of type E8 we set R=ℤ[1/30]. Given a linear function χ on 𝔤 we denote by Uχ(𝔤) the reduced enveloping algebra of 𝔤 associated with χ. By the Kac-Weisfeiler conjecture (now a theorem), any Uχ(𝔤)-module has dimension divisible by pd(χ) where 2d(χ) is the dimension of the coadjoint G-orbit of χ. In my talk, based on a joint work with Lewis Topley, I'll discuss a natural question raised in the 1990s by Kac, Humphreys and myself and explain that for any χ ∈ 𝔤* the reduced enveloping algebra Uχ(𝔤) has an irreducible module of dimension pd(χ). Forms of finite W-algebras over the ring R and their reductions modulo good primes play a crucial role in our arguments. We also use some recent results on multiplicty-free primitive ideals of U(𝔤𝒞) associated with the rigid nilpotent orbits in complex simple Lie algebras 𝔤𝒞.

Scott Larson: Small Resolutions of Closures of K-Orbits in Flag Varieties II

21st September, 2020

The geometry of closures of K-orbits in the flag variety governs key properties in representation theory of real reductive groups. For example, Kazhdan-Lusztig-Vogan polynomials and characteristic cycles of Harish-Chandra modules are of current interest. We recall how small resolutions have been used to compute these invariants, describe fibres of the resolutions from last week, and describe more small resolutions for the real reductive groups Sp2n(ℝ), U(p,q), and complex groups. Along the way we consider an application to Schubert varieties.

Chris Bowman: Tautological p-Kazhdan-Lusztig Theory for cyclotomic Hecke algebras

15th September, 2020

We discuss a new explicit isomorphism between (truncations of) quiver Hecke algebras and Elias-Williamson’s diagrammatic endomorphism algebras of Bott-Samelson bimodules. This allows us to deduce that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are tautologically equal to the associated p-Kazhdan-Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. This allows us to give an elementary and explicit proof of the main theorem of Riche-Williamson’s recent monograph and extend their categorical equivalence to cyclotomic Hecke algebras, thus solving Libedinsky-Plaza’s categorical blob conjecture.

Scott Larson: Small Resolutions of Closures of K-Orbits in Flag Varieties I

14th September, 2020

The geometry of closures of K-orbits in the flag variety governs key properties in representation theory of real reductive groups. For example, Kazhdan-Lusztig-Vogan polynomials and characteristic cycles of Harish-Chandra modules are of current interest but difficult to compute. Barbasch-Evens constructed resolutions for K-orbits on grassmannian flag varieties and found some small resolutions. We do the same thing for isotropic flag varieties of the symplectic group, where K=GLn. This leads us to describe natural resolutions for K-orbits, generalizing many constructions found in the literature.

Beth Romano: Vinberg theory and related invariant theory

28th August, 2020

An LMS online lecture course in Vinberg theory.

In recent years, Vinberg theory of graded Lie algebras has become relevant in many areas of number theory, from arithmetic statistics (e.g., in the work of Romano-Thorne) to the local Langlands correspondence (e.g., in the work of Reeder-Yu). These lectures will provide the algebraic background for number theory students to engage with research involving graded Lie algebras. We'll start by discussing some of the relevant aspects of the invariant theory of Lie algebras, including the Chevalley restriction theorem and the pioneering work of Kostant on invariant rings. We'll then define graded Lie algebras and look at the graded analogues of these theorems, based on work of Vinberg. Time permitting, we'll look at Slodowy slices and applications to families of algebraic curves. These lectures should give number theory students sufficient background to read, for example, Thorne's paper Vinberg's representations and arithmetic invariant theory and other related papers. But the lectures will also be a useful introduction to some beautiful aspects of Lie theory for students in algebra and representation theory. I'll assume students have some knowledge of Lie algebras, but I will review relevant background and provide examples throughout the lectures.

Dražen Adamović: On logarithmic and Whittaker modules for affine vertex algebras

6th August, 2020

Simple affine vertex algebras at admissible levels are semi-simple in the category O, but beyond the category O they contain interesting categories of representations with many new research challenges. We will first present our explicit lattice realizations of simple affine VOA Lk(𝔰𝔩2) at arbitrary admissible level k, and their modules in certain categories. Then we discuss the existence and explicit realization of logarithmic modules which appear as extensions of weight modules. The next natural task is to include Whittaker modules in the representation category. Although Whittaker modules are constructed using standard Lie-theoretic constructions, we will show that in order to understand the structure of affine Whittaker modules, one needs to apply vertex-algebraic techniques. We present explicit realization of Whittaker modules for some vertex algebras. We will discuss our recent efforts to generalize this realization in higher-rank cases.

Vera Serganova: The celebrated Jacobson-Morozov theorem for Lie superalgebras via semisimplification functor for tensor categories

11th June, 2020

The famous Jacobson-Morozov theorem claims that every nilpotent element of a semisimple Lie algebra 𝔤 can be embedded into an 𝔰𝔩2-triple inside 𝔤. Let 𝔤 be a Lie superalgebra with reductive even part and x be an odd element of 𝔤 with non-zero nilpotent [x,x]. We give necessary and sufficient condition when x can be embedded in 𝔬𝔰𝔭(1|2) inside 𝔤. The proof follows the approach of Etingof and Ostrik and involves semisimplification functor for tensor categories. Next, we will show that for every odd x in 𝔤 we can construct a symmetric monoidal functor between categories of representations of certain superalgebras. We discuss some properties of these functors and applications of them to representation theory of superalgebras with reductive even part. (Joint work with Inna Entova-Aizenbud).

James East: Presentations for tensor categories

15th May, 2020

Many well-known families of groups and semigroups have natural categorical analogues: e.g., full transformation categories, symmetric inverse categories, as well as categories of partitions, Brauer/Temperley-Lieb diagrams, braids and vines. This talk discusses presentations (by generators and relations) for such categories, utilising additional tensor/monoidal operations. The methods are quite general, and apply to a wide class of (strict) tensor categories with one-sided units.