## The RepNet Virtual Seminar

**Organisers:**

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

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

Wednesdays 16:00-17:00 GMT from 2nd September

To sign up to attend this seminar series please register here. The 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 see here

### Future seminars

**October 28th**, Amit Hazi, TBA

**November 4th**, Alistair Craw, TBA

**November 11th**, TBA

**November 18th**, TBA

**November 25th**, Lucas Mason-Brown

**December 2nd**, Eleonore Faber

**December 9th**, Xuhua He **THIS WEEK ONLY: SEMINAR BEGINS AT 14:00 GMT**

**December 16th**, TBA

### Previous seminars

23 June 2020

**Jacinta Torres**

*A positive combinatorial formula for symplectic Kostka-Foulkes polynomials I: Rows *

Abstract: 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)

30 June 2020

**Meinolf Geck**

*What is bad about bad primes?*

Abstract: 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).

7 July 2020

** Laura Rider**

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

Abstract: 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.

14 July 2020

** Andrew Snowden**

*The representation theory of Brauer categories*

Abstract: 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.

21 July 2020

**Gunter Malle**

*What Weyl groups know*

Abstract: 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.

28 July 2020

**Leo Patimo**

*Extending Schubert calculus to intersection cohomology*

Abstract: 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

04 August 2020

**Laura Colmenarejo**

*An insertion algorithm for diagram algebras*

Abstract: 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.

02 September 2020

** Martina Lanini **

*Torus actions on cyclic quiver Grassmannians*

Abstract: 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.

Wednesday 9th September Speaker: Eirini Chavli (Universität Stuttgart) Title: Real properties of generic Hecke algebras Abstract: 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.

16 September 2020

**Nicolle Gonzalez**

*A skein theoretic Carlsson-Mellit algebra.*

Abstract: 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.

23 September 2020

**Francesco Sala**

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

Abstract: 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.

30 September

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

Title: *Simple transitive 2-representations of Soergel bimodules for finite Coxeter type in characteristic zero*

Abstract: 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.

October 7th

**Julian Kuelshammer**

Title: *Filtered categories via ring extensions*

Abstract: 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.

**October 14th**, Bea Schumann

Title: *Parametrizations of canonical bases of representations of algebraic
groups via cluster duality*

Abstract: 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.

**October 21st**, Dave Benson,

Title:* Some exotic tensor categories in prime characteristic*

Abstract: 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).

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