# Abstracts

### Upcoming talks

Speaker: Radha Kessar

Title: The principal block of a Z_l-spets and Yokonuma type algebras.

Abstract: I will report on joint work with Gunter Malle and Jason Semeraro. Motivated by various conjectures in finite group theory, we have introduced the notion of the principal block of a Z_l spets. The definition uses input from spetsial character theory as well as from the theory of l-compact groups. In the case that the underlying complex reflection group of the Z_l spets is a Weyl group, we recover the data of the irreducible characters of the principal l-block of the relevant finite group of Lie type. Many features of the modular representation theory of finite groups carry over to the Z_l-spets setting. For instance, one can formulate analogues of the local-global counting conjectures for the principal block of a Z_l-spets. In very recent work, we have also formulated conjectures concerning the dimension of the principal block of a

Z_l-spets. We show that these dimension conjectures hold in many cases-the proof goes via a new Yokonuma type algebra.

### Past talks

Jacinta Torres (23rd June). A positive combinatorial formula for symplectic Kostka-Foulkes polynomials I: Rows

Fix a simple Lie algebra over the complex numbers. Kostka–Foulkes polynomials are defined for two dominant integral weights as the transition coefficients between two important bases of the ring of symmetric functions: Hall–Littlewood polynomials and Weyl characters. Due to their interpretation as affine Kazhdan-Lusztig polynomials, they are known to have non-negative integer coefficients. However, a closed combinatorial formula is yet to be found outside of type An, where the celebrated charge formula of Lascoux-Schützenberger stands alone. In type Cn, Lecouvey conjectured a charge formula in terms of symplectic cocyclage and Kashiwara-Nakashima tableaux. We reformulate and prove his conjecture for rows of arbitrary weight, and present an algorithm which we believe could well lead to a proof of the conjecture in general.

This is joint work with Maciej Dołęga and Thomas Gerber. (arXiv:1911.06732)

Meinolf Geck (30th June). What is bad about bad primes?

Let G be a connected reductive algebraic group defined over a finite field with q elements. In the 1980’s, Kawanaka introduced generalised Gelfand-Graev representations of the finite group G(Fq), assuming that q is not a power of a “bad” prime for G. These representations have turned out to be extremely useful in various contexts. In an attempt to extend Kawanaka’s construction to the “bad” prime case, we proposed a new characterisation of Lusztig’s concept of special unipotent classes of G (which is now a theorem).

Laura Rider (7th July). 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.

Andrew Snowden (14th July). The representation theory of Brauer categories

Brauer introduced a family of algebras, now called Brauer algebras, in an effort to extend Schur-Weyl duality to the orthogonal groups. This family of algebras can be assembled into a single object: the Brauer category. In this talk, I will describe various aspects of the representation theory of this category (and some of its cousins). It can be viewed both from the point of view of representation theory and commutative algebra, and connects to many other topics, such as supergroups and Deligne's interpolation categories. This is joint work with Steven Sam.

Gunter Malle (21st July). 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.

Leonardo Patimo (28th July). Extending Schubert calculus to intersection cohomology

The Schubert basis is a distinguished basis of the cohomology of a Schubert variety which contains rich information about the ring structure of the cohomology. When working with the intersection cohomology, we do not have a Schubert basis in general, and in fact understanding the intersection cohomology of a Schubert variety can be much more difficult. However, one may often exploit the knowledge of the corresponding Kazhdan-Lusztig polynomials to produce new bases in intersection cohomology which extend the original Schubert basis. In this talk we will see two different situations where this is possible, although the solutions have quite different flavours: Schubert varieties in Grassmannians and (jt. with N. Libedinsky) Schubert varieties for the affine Weyl group $\widetilde{A}_2$.

Laura Colmenarejo (4th August). 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.

Martina Lanini (September 2). Torus actions on cyclic quiver Grassmannians

I will report on recent joint work with Alexander Puetz, where we define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. These quiver Grassmannians, equipped with such torus actions, are equivariantly formal spaces, and the corresponding moment graphs can be combinatorially described and exploited to compute equivariant cohomology. Our construction generalises the very much investigated (maximal) torus actions on type A flag varieties.

Eirini Chavli (September 9). 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.

Nicolle Gonzalez (Sept. 16), A skein theoretic Carlsson-Mellit algebra

The Carlsson-Mellit algebra, or $A_{q,t}$ algebra, originally arose in the proof of the celebrated Shuffle conjecture, which gives a combinatorial formula for the Frobenius character of the space of diagonal harmonics. This algebra, built from Hecke algebra generators and a family of raising and lowering operators, has a particularly interesting representation, known as the polynomial representation, on which its action is given by complicated plethystic operations. In this talk I will discuss how this algebra (specialized at $t=q^{-1}$) and its polynomial representation can be formulated skein theoretically as certain braid diagrams on a thickened annulus. Using the recent construction of Gorsky-Hogancamp-Wedrick of the derived trace of the Soergel category, we lift the skein formulation to a categorification of the polynomial representation of $A_{q,t}$. This is joint work with Matt Hogancamp.

Francesco Sala (Sept. 23), 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.

Vanessa Miemietz (Sept. 30), Simple transitive 2-representations of Soergel bimodules for finite Coxeter type in characteristic zero

I will explain how to relate simple transitive 2-representations of Soergel bimodules for finite Coxeter type in characteristic zero to 2-representations of certain fusion categories, which are, for the most part, well understood.

Julian Kuelshammer (October 7th). 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.

Bea Schumann, (October 14th) Parametrizations of canonical bases of representations of algebraic groups via cluster duality

Let G be a simple, simply connected algebraic group over the complex numbers. The algebra of regular functions on the base affine space of G splits into the multiplicity free direct sum of all finite dimensional irreducible G-representations. Gross-Hacking-Keel-Kontsevich constructed, under some combinatorial conditions, a basis of this algebra which is compatible with this decomposition. This basis is provided by a duality theorem using the cluster algebra structure of the base affine space and comes naturally with nice parametrizations by polytopes. We explain why Gross-Hacking-Keel-Kontsevich’s conditions are satisfied and analyse the combinatorics of the parametrizations. This is based on joint work with V. Genz and G. Koshevoy.

Dave Benson, 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).

Amit Hazi, Path isomorphisms between quiver Hecke and diagrammatic Bott-Samelson endomorphism algebras

We construct an explicit isomorphism between certain truncations of quiver Hecke algebras and Elias-Williamson's diagrammatic endomorphism algebras of Bott-Samelson bimodules. As a corollary, we deduce that the decomposition numbers of these algebras (including as examples the symmetric groups and generalised blob algebras) are equal to the associated p-Kazhdan-Lusztig polynomials, provided that the characteristic is greater than the Coxeter number. In the special case of the symmetric group this gives an elementary and more explicit proof of the tilting character formula of Riche-Williamson's recent monograph. Based on joint work with Chris Bowman and Anton Cox.

Alastair Craw, 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.

Alexandre Minets, KLR and Schur algebras for curves and semi-cuspidal representations

Given a smooth curve C, we define and study analogues of quiver Hecke and Schur algebras, where quiver representations are replaced by torsion sheaves on C. When C is a projective line, we deduce a description of the category of imaginary semi-cuspidal representations of type A_1^(1) quiver Hecke algebra in positive characteristic, thus answering a question of Kleshchev. This is a joint work with Ruslan Maksimau.

Andrea Appel, 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.

Lucas Mason-Brown (25 Nov), What is a unipotent representation?

Eleonore Faber (2nd December). McKay quivers and the McKay correspondence for some complex reflection groups

Finite complex reflection groups were classified by Shephard and Todd: up to finitely many exceptions they are the groups G(r,p,n). In this talk we give a combinatorial description of the McKay quivers of these groups. Further we will comment on a McKay correspondence for complex reflection groups. This is joint work with R.-O. Buchweitz, C. Ingalls, and M. Lewis. (2nd December)

Xuhua He (9th December). Flag varieties for semifields

Julia Pevtsova (16 December). 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.

Eoghan McDowell: The image of the Specht module under the inverse Schur functor

Abstract: The Schur functor and its inverses give an important connection between the representation theories of the symmetric group and the general linear group. Kleschev and Nakano proved in 2001 that when the characteristic of the field is at least 5, the image of the Specht module under the inverse Schur functor is isomorphic to the dual Weyl module. In this talk I will address what happens in characteristics 2 and 3: in characteristic 3, the isomorphism holds, and I will give an elementary proof of this fact which covers also all characteristics other than 2; in characteristic 2, the isomorphism does not always hold, and I will classify for which Specht modules it does.

Angela Tabiri (20 January): 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.

Asilata Bapat (January 27). A Thurston compactification of Bridgeland stability space.

The space of Bridgeland stability conditions on a triangulated category is a complex manifold. We propose a compactification of the stability space via a continuous map to an infinite projective space. Under suitable conditions, we conjecture that the compactification is a real manifold with boundary, on which the action of the autoequivalence group of the category extends continuously. We focus on 2-Calabi--Yau categories associated to quivers, and prove our conjectures in the A2 and affine A1 cases. This is joint work with Anand Deopurkar and Anthony Licata.

Inna Entova-Aizenbud (February 3). 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.

Daniel K. Nakano (February 10). A new Lie theory for simple Lie superalgebras

José Simental (February 17). 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.

Simon Riche (24th February). Hecke action on the principal block of reductive algebraic groups

I will explain a joint work with Roman Bezrukavnikov where we show that wall crossing functors define an action of the Hecke category on the principal block of the category of representations of a reductive algebraic group. This fact had been conjectured in a joint work with G. Williamson (and previously proved in type A by other methods); it gives a very concrete incarnation of the idea that "the affine Weyl group controls the category of representations," and has as a consequence character formulas for tilting and simple modules.

Pedro Vaz (3rd March), Schur-Weyl duality, Verma modules, and categorification

I will explain my joint work with Abel Lacabanne on Schur-Weyl duality involving parabolic Verma modules. Several well-known algebras, like Ariki-Koike algebras, and (generalized) blob algebras for example, appear naturally as particular cases of our main result. In the second part of the talk I will briefly review the program of categorification Verma modules and explain how it is used to categorify the blob algebra of Martin and Saleur (this is joint work with Grégoire Naisse and Abel Lacabanne).

Jiuzu Hong (10th March), 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.

Peter Tingley (17th March), A quiver approach to root multiplicities

We present combinatorial upper bounds on dimensions of certain imaginary root spaces for symmetric Kac-Moody algebras. These come from the realization of the corresponding infinity-crystal using quiver varieties. The framework is general, but we only work out specifics in rank two and certain rank 3 cases. There we give explicit bounds. These turn out to be quite accurate, and in many cases exact, even for some fairly large roots. Perhaps surprisingly, things are cleanest and most accurate in rank 2. This includes joint work with Colin Williams and with Patrick Chan.

Charlotte Chan (24th March), 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.

Karina Batistelli (March 31, 2021), Kazhdan-Lusztig polynomials for B̃2

Nicolas Libedinsky (May 5, 2021), Precanonical bases on affine Hecke algebras.

Lewis Topley (May 12, 2021), Representations of finite W-algebras.

A finite W-algebra is a non-commutative algebra which plays a key role in the representation theory of complex semisimple Lie algebras. It is constructed by quantum Hamiltonian reduction, and it gives a filtered quantisation of a transverse slice to a nilpotent orbit. Its representations have a lot to do with the geometry of the orbit closure and I will begin this talk by surveying some of the landmark results in this area. In the second part of the talk I will describe some of the techniques in my recent paper, in which I describe the variety of one dimensional representations associated to classical Lie algebras.

Robert Spencer (May 19, 2021). 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.

Henning Krause (May 26). Local versus global representations of finite groups

We consider the stable category of modular representations of a finite group. There are local cohomology functors which are parametrised by the prime ideals of the cohomology ring. These functors provide categories of local representations that are supported at a single prime. The talk will focus on these local categories. They are tensor triangulated and it turns out that compact and dualising objects do not coincide, in contrast to the category of global representations. The talk presents recent progress from an ongoing collaboration with Dave Benson, Srikanth Iyengar, and Julia Pevtsova.

Jon Brundan, 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.

Ivan Losev (9 June). *Springer, Procesi and Cherednik*

The talk is based on a joint work with Pablo Boixeda Alvarez, arXiv:2104.09543. We study equivariant Borel-Moore homology of certain affine Springer fibers and relate them to global sections of suitable vector bundles arising from Procesi bundles on Q-factorial terminalizations of symplectic quotient singularities. This relation should give some information on the center of the principal block of the small quantum group. Our main technique is based on studying bimodules over Cherednik algebras.

Mee Seong Im (16 June). *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.

Hankyung Ko (23 June). *Monoidal actions of the Hecke category and Kostant's problem*

Let g be a semisimple complex Lie algebra and let M be a g-module. Consider A(M), the space of linear endomorphisms on M on which the adjoint action of g is finite. A classical question of Kostant is: for which simple module M is the canonical map from U(g) to A(M) surjective? I will reformulate this problem using the language of monoidal (or 2-)categories. If M belongs to the BGG category O, then the answer to the problem is equivalent to equivalence of certain categorical actions of the Hecke category, and is determined by decomposing the action of translation functors on M. This leads to a conjectural answer to Kostant's problem in terms of the Kazhdan-Lusztig basis for the Hecke algebra. This is a joint work with Walter Mazorchuk and Rafael Mrden.

Mina Aganagic (July 14). 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.