*Jan 27 2021*

10:00 - 11:00

#### Organisers:

**Chris Bowman,**University of Kent**Kevin McGerty,**University of Oxford**Emily Norton,**TU Kaiserslautern**Tomasz Przezdziecki,**University of Edinburgh**Ulrich Thiel,**TU Kaiserslautern

Registration will close at 12.00 on the day of the seminar and a link to the meeting will be sent to participants then. The meetings will run in Zoom.

The organisers have their own website which you can access here.

### Future seminars

27th January 2021

**Asilata Bapat**

THIS WEEK ONLY: SEMINAR BEGINS AT 10:00 GMT

3rd February 2021

**Inna Entova-Aizenbud **

17th February 2021

**Jose Simental **

### Previous seminars

23rd June 2020

**Jacinta Torres - 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)

30th June 2020

**Meinolf Geck - 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).

7th July 2020

**Laura Rider - 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.

14th July 2020

**Andrew Snowden - 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 super groups and Deligne's interpolation categories. This is joint work with Steven Sam.

21th July 2020

**Gunter Malle - 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.

28th July 2020

**Leo Patimo - 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 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

4th August 2020

**Laura Colmenarejo - 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.

2nd September 2020

**Martina Lanini - 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.

9th September 2020

**Eirini Chavli (Universität Stuttgart) - 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.

16th September 2020

**Nicolle Gonzalez - 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.

23rd September 2020

**Francesco Sala - 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.

30th September 2020

**Vanessa Miemietz, (the University of East Anglia) - 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.

7th October 2020

**Julian Kuelshammer - 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.

14th October 2020

**Bea Schumann - 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.

21st October 2020

**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).
- View the lecture slides here

28th October 2020

**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.

4th November 2020

**Alistair Craw - 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.

11th November 2020

**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.

18th November 2020

**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.

25th November 2020

**Lucas Mason-Brown - What is a unipotent representation**

- View the abstract here

2nd December 2020

**Eleonore Faber**

9th December 2020

**Xuhua He**

16th December 2020

**Julia Pevtsova - 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.

13th January 2021

**Eoghan McDowell - 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.

20th January 2021

**Angela Tabiri - 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.

**This seminar series is supported as part of the ICMS Online Mathematical Sciences Seminars.**