Centralizer of a regular unipotent element and perverse sheaves on the affine flag variety

In this talk, I will give a geometric description of the category of representations of the centralizer of a regular unipotent element in a reductive algebraic group in terms of perverse sheaves on the Langlands dual affine flag variety. This is joint work with R. Bezrukavnikov and S. Riche.

What Weyl groups know

Weyl groups lie at the core of various quite very different mathematical structures. They not only control much of the behaviour of these objects they also allow us to transfer notions from one setting to another. In the talk I will try to motivate and explain work in progress with Radha Kessar and Jason Semeraro on how and why the Alperin and Robinson weight conjectures from modular representation theory of finite groups also do make sense (and continue to hold) for $ell$-compact groups from algebraic topology. If technology permits, this will be a blackboard talk.

An insertion algorithm for diagram algebras

We generalize the Robinson-Schensted-Knuth algorithm to the insertion of two-row arrays of multisets. This generalization leads to an algorithm from partition diagrams to pairs of a standard tableau and a standard multiset tableau of the same shape, which has the remarkable property that it is well-behaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representation-theoretic results of Halverson and Jacobson. This is joint work with Rosa Orellana, Franco Saliola, Anne Schilling, and Mike Zabrocki.

Real properties of generic Hecke algebras

Iwahori Hecke algebras associated with real reflection groups appear in the study of finite reductive groups. In 1998 Broué, Malle and Rouquier generalised in a natural way the definition of these algebras to the complex case, known now as generic Hecke algebras. However, some basic properties of the real case were conjectured for generic Hecke algebras. In this talk we will talk about these conjectures and their state of the art.

Two-dimensional cohomological Hall algebras of curves and surfaces, and their categorification

In the present talk, I will broadly introduce two-cohomological Hall algebras of curves and surfaces and discuss their categorification. In the second part of the talk, I will discuss in detail the example of a cohomological Hall algebra when the surface is the minimal resolution of a type A singularity. This is based on papers with Diaconescu, Schiffmann, and Porta.

Filtered categories via ring extensions

Important examples of exact categories are categories of objects filtered by a collection of special objects. In Lie theory, one of the prototypical instances is the subcategory of BGG category O of modules filtered by Verma modules. In this case, the Verma modules are induced from the Borel subalgebra. More generally, in 2014 together with Steffen Koenig and Sergiy Ovsienko we showed that categories of filtered modules for quasi-hereditary algebras can be realised as induced modules for a ring extension. In this talk, we will give an alternative approach to this theorem and discuss uniqueness of the ring extension. This is based on joint work with Tomasz Brzezinski, Steffen Koenig and Vanessa Miemietz.

Some exotic tensor categories in prime characteristic

I shall talk about some recent joint work with Etingof and Ostrik, producing some new incompressible symmetric tensor categories in prime characteristic and explain some of their properties and potential role in the theory. The input for the construction is the theory of tilting modules for SL(2).

Gale duality and the linearisation map for quiver moduli

I'll talk about the linearisation map for fine quiver moduli spaces. These spaces are constructed as geometric invariant theory quotients X//G, and the linearisation map assigns to each character of G the corresponding line bundle on the quotient X//G obtained by descent. I'll present natural geometric conditions that guarantee this map to the Picard group is surjective and I'll describe the geometry that is encoded in the Gale dual map. The key point of the talk is to examine two matrices - one for the linearisation map and the other for its Gale dual - and to show that two rival interpretations of `Reid's recipe' for a finite subgroup of SL(3,C) actually encode the same information.

Reflection symmetries arising from quantum Kac-Moody algebras

I will report on recent and ongoing joint work with Bart Vlaar aimed at the construction of k-matrices (that is, solution of twisted reflection equations) for category O integrable representations of quantum Kac-Moody algebras. In the affine case, our construction is conjectured to extend to the category of finite-dimensional representation of quantum loop algebras, producing a parameter-dependent meromorphic operator satisfying the spectral reflection equation

Cohomology of finite dimensional Hopf algebras

In 2004 Etingof and Ostrik stated in print a conjecture which had existed in a folkloric form for at least two decades before that: "the cohomology ring of a finite tensor category is finitely generated". The first evidence of the conjecture appears in the work of Golod, Venkov and Evens in 1959-1961 who showed that the cohomology of a finite group with mod p coefficients is finitely generated. Since then finite generation of cohomology has been shown for many different representation categories, such as the ones for modular Lie algebras, small quantum groups, Lie superalgebras, finite group schemes, and Nichols algebras of diagonal type. The general case of the conjecture remains wide open. Despite a purely algebraic nature of the conjecture it has profound geometrically flavored consequences in the field of triangular geometry. I'll give an overview of the history and the current state of the problem, and touch upon some geometric applications which are the major driving forces for the recent interest in the finite generation conjecture. Based on joint work with N. Andruskiewitsch, I Angiono, S. Witherspoon, and C. Negron

Algebraic Structures in Group Theoretical Fusion Categories

It was shown by Ostrik (2003) and Natale (2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We generalize this result by constructing explicit Morita equivalence class representatives of indecomposable, separable algebras in group-theoretical fusion categories. This is achieved by providing the `free functor' Φ from a pointed fusion category to a group-theoretical fusion category with a monoidal structure. Our algebras of interest are then constructed as the image of twisted group algebras under Φ. We also show that twisted group algebras admit the structure of Frobenius algebras in a pointed fusion category, and we establish a Frobenius monoidal structure on Φ as well. As a consequence, our algebras are Frobenius algebras in a group-theoretical fusion category, and like twisted group algebras in the pointed case, they also enjoy several good algebraic properties

Superdimensions of simple modules for the periplectic Lie superalgebra

The periplectic Lie superalgebra is one of the classical Lie superalgebras arising from V. Kac's classification of simple Lie superalgebras. It has a rich finite-dimensional representation theory. I will describe a simple explicit combinatorial formula for the superdimension of a simple (integrable) finite-dimensional p(n)-module, and the tools used to obtain this formula. This is joint work with V. Serganova.

Braid varieties and Algebraic weaves

To a positive braid we associate an affine algebraic variety that we call the braid variety. I will define these varieties and explain some of their basic properties, as well as their connections to other varieties such as augmentation varieties for Legendrian links and open Bott-Samelson varieties. To study braid varieties we develop algebraic weaves, a diagrammatic calculus for correspondences between them that has many similarities to Soergel calculus but differs from it in key aspects. This is joint work with Roger Casals, Eugene Gorsky and Mikhail Gorsky.

Nilpotent varieties in symmetric spaces and twisted affine Schubert varieties

I will talk about a connection between nilpotent varieties in symmetric spaces and Schubert varieties in twisted affine Grassmannian. This extends some results of Lusztig and Achar-Henderson in a twisted setting. This talk will be based on the joint work with Korkeat Korkeathikhun.

Flag varieties and representations of p-adic groups

Deligne--Lusztig varieties are subvarieties of flag varieties whose cohomology encodes the representations of reductive groups over finite fields. These give rise to so-called "depth-zero" supercuspidal representations of p-adic groups. In this talk, we discuss geometric constructions of positive depth supercuspidal representations and the implications of such realizations towards the Langlands program. This is partially based on joint work with Alexander Ivanov and joint work with Masao Oi.

Pre-canonical bases on affine Hecke algebras

The Modular Temperley-Lieb Algebra.

The Temperley-Lieb algebra is a diagrammatic algebra - defined on a basis of "string diagrams" with multiplication given by "joining the diagrams together".  It first arose as an algebra of operators in statistical mechanics but quickly found application in knot theory (where Jones used it to define his famed polynomial) and the representation theory of sl_2.  From the outset, the representation theory of the Temperley-Lieb algebra itself has been of interest from a physics viewpoint and in characteristic zero it is well understood.  In this talk we will explore the representation theory over mixed characteristic (i.e. over positive characteristic fields and specialised at a root of unity).  This gentle introduction will take the listener through the beautifully fractal-like structure of the algebras and their cell modules with plenty of examples.

Heisenberg actions on Abelian categories

I will talk about my joint work with Alistair Savage and Ben Webster which develops a general framework for studying actions of the infinite-dimensional Heisenberg Lie algebra on locally finite Abelian categories. The first example comes from Khovanov’s Heisenberg category acting on representations of symmetric groups. In fact, most of the important Abelian categories appearing in “type A” representation theory admit an action by an appropriately defined Heisenberg category, and our approach unifies the study of all of these examples. Time permitting, I hope also to explain a new application of the Heisenberg graphical calculus to another interesting combinatorial category—the partition category.

Irreducible components of two-row Springer fibers for all classical types

I will give an explicit description of the irreducible components of two-row Springer fibers for all classical types using cup diagrams. Cup diagrams can be used to label the irreducible components of two-row Springer fibers. Given a cup diagram, we explicitly write down all flags contained in the component associated with the cup diagram. This generalizes results by Stroppel–Webster and Fung to all classical types. This is joint work with Chun-Ju Lai and Arik Wilbert.

Please note this seminar was NOT recorded

On the computation of decomposition numbers of the symmetric group.

The most important open problem in the modular representation theory of the symmetric group is finding the multiplicity of the simple modules as composition factors of the Specht modules. In characteristic 0 the Specht modules are just the simple modules of the symmetric group algebra, but in positive characteristic they may no longer be simple. We will survey the rich interplay between representation theory and combinatorics of integer partitions, review a number of results in the literature which allow us to compute composition series for certain infinite families of Specht modules from a finite subsetof them, and discuss the extension of these techniques to other Specht modules.

Knot homologies from mirror symmetry

Khovanov showed, more than 20 years ago, that the Jones polynomial emerges as an Euler characteristic of a homology theory. The knot categorification problem is to find a uniform description of this theory, for all gauge groups, which originates from physics, or from geometry. I will describe two solutions to this problem, which are related by a version of homological mirror symmetry.