## 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

Tuesdays 16.00-17.00 BST beginning Tuesday 23 June

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

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**

28 July 2020 ** Leo Patimo**

4 August 2020 ** Laura Colmenarejo**

### 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)

Tuesday 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).

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