
Chris Bowman, University of Kent

Kevin McGerty, University of Oxford

Emily Norton, TU Kaiserslautern

Tomasz Przezdziecki, University of Edinburgh

Ulrich Thiel, TU Kaiserslautern
Past Events:
A positive combinatorial formula for symplectic KostkaFoulkes polynomials I: Rows
Fix a simple Lie algebra over the complex numbers. KostkaFoulkes polynomials are defined for two dominant integral weights as the transition coefficients between two important bases of the ring of symmetric functions: HallLittlewood polynomials and Weyl characters. Due to their interpretation as affine KazhdanLusztig polynomials, they are known to have nonnegative integer coefficients. However, a closed combinatorial formula is yet to be found outside of type An, where the celebrated charge formula of LascouxSchützenberger stands alone. In type Cn, Lecouvey conjectured a charge formula in terms of symplectic cocyclage and KashiwaraNakashima 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)
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 GelfandGraev 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).
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.
The representation theory of Brauer categories
Brauer introduced a family of algebras, now called Brauer algebras, in an effort to extend SchurWeyl 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 super groups and Deligne's interpolation categories. This is joint work with Steven Sam.
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.
Extending Schubert calculus to intersection cohomology
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 KazhdanLusztig 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
An insertion algorithm for diagram algebras
We generalize the RobinsonSchenstedKnuth algorithm to the insertion of tworow 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 wellbehaved with respect to restricting a representation to a subalgebra. This insertion algorithm matches recent representationtheoretic results of Halverson and Jacobson. This is joint work with Rosa Orellana, Franco Saliola, Anne Schilling, and Mike Zabrocki.
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.
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.
A skein theoretic CarlssonMellit algebra
The CarlssonMellit 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 GorskyHogancampWedrick 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.
Twodimensional cohomological Hall algebras of curves and surfaces, and their categorification
In the present talk, I will broadly introduce twocohomological 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.
Simple transitive 2representations of Soergel bimodules for finite Coxeter type in characteristic zero
I will explain how to relate simple transitive 2representations of Soergel bimodules for finite Coxeter type in characteristic zero to 2representations of certain fusion categories, which are, for the most part, well understood.
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 quasihereditary 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.
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 Grepresentations. GrossHackingKeelKontsevich 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 GrossHackingKeelKontsevich's conditions are satisfied and analyse the combinatorics of the parametrizations. This is based on joint work with V. Genz and G. Koshevoy.
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).
Path isomorphisms between quiver Hecke and diagrammatic BottSamelson endomorphism algebras
We construct an explicit isomorphism between certain truncations of quiver Hecke algebras and EliasWilliamson's diagrammatic endomorphism algebras of BottSamelson 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 pKazhdanLusztig 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 RicheWilliamson's recent monograph. Based on joint work with Chris Bowman and Anton Cox.
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.
KLR and Schur algebras for curves and semicuspidal 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 semicuspidal 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.
Reflection symmetries arising from quantum KacMoody algebras
I will report on recent and ongoing joint work with Bart Vlaar aimed at the construction of kmatrices (that is, solution of twisted reflection equations) for category O integrable representations of quantum KacMoody algebras. In the affine case, our construction is conjectured to extend to the category of finitedimensional representation of quantum loop algebras, producing a parameterdependent meromorphic operator satisfying the spectral reflection equation
What is a unipotent representation
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 19591961 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
The image of the Specht module under the inverse Schur functor
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.
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 grouptheoretical fusion categories. This is achieved by providing the `free functor' Φ from a pointed fusion category to a grouptheoretical 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 grouptheoretical fusion category, and like twisted group algebras in the pointed case, they also enjoy several good algebraic properties
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 2CalabiYau 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.
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 finitedimensional representation theory. I will describe a simple explicit combinatorial formula for the superdimension of a simple (integrable) finitedimensional p(n)module, and the tools used to obtain this formula. This is joint work with V. Serganova.
A new Lie theory for simple Lie superalgebras
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 BottSamelson 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.
SchurWeyl duality, Verma modules, and categorification
I will explain my joint work with Abel Lacabanne on SchurWeyl duality involving parabolic Verma modules. Several wellknown algebras, like ArikiKoike 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
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 AcharHenderson in a twisted setting. This talk will be based on the joint work with Korkeat Korkeathikhun.
A quiver variety approach to root multiplicities
Flag varieties and representations of padic groups
DeligneLusztig varieties are subvarieties of flag varieties whose cohomology encodes the representations of reductive groups over finite fields. These give rise to socalled "depthzero" supercuspidal representations of padic 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.
KazhdanLusztig polynomials for B̃2
Precanonical bases on affine Hecke algebras
Representations of finite Walgebras
A finite Walgebra is a noncommutative 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.
The Modular TemperleyLieb Algebra.
The TemperleyLieb 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 TemperleyLieb 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 fractallike structure of the algebras and their cell modules with plenty of examples.
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.
This Lecture was NOT recorded
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 infinitedimensional 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.
Springer, Procesi and Cherednik
The talk is based on a joint work with Pablo Boixeda Alvarez, arXiv:2104.09543. We study equivariant BorelMoore homology of certain affine Springer fibers and relate them to global sections of suitable vector bundles arising from Procesi bundles on Qfactorial 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.
Irreducible components of tworow Springer fibers for all classical types
I will give an explicit description of the irreducible components of tworow Springer fibers for all classical types using cup diagrams. Cup diagrams can be used to label the irreducible components of tworow 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 ChunJu Lai and Arik Wilbert.
Please note this seminar was NOT recorded
Monoidal actions of the Hecke category and Kostant's problem
Let g be a semisimple complex Lie algebra and let M be a gmodule. 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 KazhdanLusztig basis for the Hecke algebra. This is a joint work with Walter Mazorchuk and Rafael Mrden.
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.
Fusion rules for SL2
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.
The Affine Partition Algebra
In this talk we present a new algebra called the affine partition algebra. It plays a similar role for the partition algebra as that of the affine degenerate Hecke algebra for the group algebra of the symmetric group. We give some motivation behind its construction and highlight some of its properties including an action which extends the SchurWeyl duality between the partition algebra and the symmetric group. We also establish a connection to the Heisenberg category by realising an endomorphism algebra of a certain object in this category as a quotient of the affine partition algebra.
The principal block of a Z_lspets and Yokonuma type algebras
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 lcompact 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 lblock of the relevant finite group of Lie type. Many features of the modular representation theory of finite groups carry over to the Z_lspets setting. For instance, one can formulate analogues of the localglobal counting conjectures for the principal block of a Z_lspets. In very recent work, we have also formulated conjectures concerning the dimension of the principal block of a Z_lspets. We show that these dimension conjectures hold in many casesthe proof goes via a new Yokonuma type algebra.