Test

Test

Dietrich Burde: Pre-Lie algebra structures on reductive Lie algebras and étale affine representations

27th February, 2023

Étale affine representations of Lie algebras and algebraic groups arise in the context of affine geometry on Lie groups, operad theory, deformation theory and Yang-Baxter equations. For reductive groups, every étale affine representation is equivalent to a linear representation and we obtain a special case of a prehomogeneous representation. Such representations have been classified by Sato and Kimura in some cases. The induced representation on the Lie algebra level gives rise to a pre-Lie algebra structure on the Lie algebra 𝔤 of G. For a Lie group G, a pre-Lie algebra structure on 𝔤 corresponds to a left-invariant affine structure on G. This refers to a well-known question by John Milnor from 1977 on the existence of complete left-invariant affine structures on solvable Lie groups. We present results on the existence of étale affine representations of reductive groups and Lie algebras and discuss a related conjecture of V. Popov concerning flattenable groups and linearizable subgroups of the affine Cremona group.

Daniel Nakano: On Tilting Modules and p-Filtrations

3rd October, 2022

In this talk, I will present background material on two conjectures formulated by Donkin at MSRI in 1990. The first conjecture is Donkin's Tilting Module Conjecture (DTilt), and the second conjecture is Donkin's p-Filtration Conjecture (DFilt). Recent progress by Kildetoft-Nakano and Sobaje has shown that there are important connections between these conjectures. In particular, Jantzen's Question posed in 1980 on the existence of Weyl p-filtrations for Weyl modules for a reductive algebraic group constitutes a central part of the new developments. I will describe how we produced infinite families of counterexamples to Jantzen's Question and Donkin's Tilting Module Conjecture. New techniques to exhibit explicit examples are provided along with methods to produce counterexamples in large rank from counterexamples in small rank. Counterexamples can be produced via our methods for all groups other than when the root system is of type An or B2. Later, I will also present a complete answer to Donkin’s Tilting Module Conjecture for rank 2 groups.

Alistair Savage: Diagratification

1st September, 2022

We will explain how one can construct diagrammatic presentations of categories of representations of Lie groups and their associated quantum groups using only a small amount of information about these categories. To illustrate the technique in concrete terms, we will focus on the exceptional Lie group of type F4.

Paul Sobaje: Two problems related to tilting modules for algebraic groups

11th April, 2022

The modular representation theory of reductive algebraic groups (general linear groups being an example of such groups) has a number of longstanding open problems. Several of these problems have conjectured resolutions that involve special modules known as indecomposable tilting modules. In this talk we will look at how tilting modules relate to two problems in particular:

   1.  lifting representations from the Lie algebra to the algebraic group, and
   2.  finding a character formula for the irreducible representations.

A good deal of background material will be provided throughout.

William Graham: Geometry of generalized Springer fibres II

7th February, 2022

Previous work constructed an analogue of the Springer resolution for the universal cover of the principal nilpotent orbit. In joint work with Precup and Russell, we showed that in type A this generalized Springer resolution is closely connected with Lusztig's generalized Springer correspondence. In this talk we discuss the geometry of the fibres of the generalized Springer resolution, and in particular, show that the fibres have an analogue of an affine paving.

William Graham: Geometry of generalized Springer fibres I

31st January, 2022

Previous work constructed an analogue of the Springer resolution for the universal cover of the principal nilpotent orbit. In joint work with Precup and Russell, we showed that in type A this generalized Springer resolution is closely connected with Lusztig's generalized Springer correspondence. In this talk we discuss the geometry of the fibres of the generalized Springer resolution, and in particular, show that the fibres have an analogue of an affine paving.

Tianyuan Xu: On Kazhdan–Lusztig cells of a-value 2

30th November, 2021

The Kazhdan–Lusztig (KL) cells of a Coxeter group are subsets of the group defined using the KL basis of the associated Iwahori–Hecke algebra. The cells of symmetric groups can be computed via the Robinson–Schensted correspondence, but for general Coxeter groups combinatorial descriptions of KL cells are largely unknown except for cells of a-value 0 or 1, where a refers to an ℕ-valued function defined by Lusztig that is constant on each cell. In this talk, we will report some recent progress on KL cells of a-value 2. In particular, we classify Coxeter groups with finitely many elements of a-value 2, and for such groups we characterize and count all cells of a-value 2 via certain posets called heaps. We will also mention some applications of these results for cell modules.

George Seelinger: Diagonal harmonics and shuffle theorems

26th October, 2021

The Shuffle Theorem, conjectured by Haglund, Haiman, Loehr, Remmel and Ulyanov, and proved by Carlsson and Mellit, describes the characteristic of the Sn-module of diagonal harmonics as a weight generating function over labelled Dyck paths under a line with slope −1. The Shuffle Theorem has been generalized in many different directions, producing a number of theorems and conjectures. We provide a generalized shuffle theorem for paths under any line with negative slope using different methods from previous proofs of the Shuffle Theorem. In particular, our proof relies on showing a 'stable' shuffle theorem in the ring of virtual GLℓ-characters. Furthermore, we use our techniques to prove the Extended Delta Conjecture, yet another generalization of the original Shuffle Conjecture.

Jiuzu Hong: Smooth Locus of Twisted Affine Schubert Varieties and Twisted Affine Demazure Modules

17th October, 2021

Let G be a special parahoric group scheme of twisted type, excluding the absolutely special case for twisted A2n. Using the methods and results of Zhu, we prove a duality theorem for general G: there is a duality between the level one twisted affine Demazure modules and function rings of certain torus fixed point subschemes in twisted affine Schubert varieties for G. Along the way, we also establish the duality theorem for untwisted E6. As a consequence, we determine the smooth
locus of any affine Schubert variety in affine Grassmannian of G, which confirms a conjecture of Haines and Richarz.

Hankyung Ko: Bruhat orders and Verma modules

28th September, 2021

The Bruhat order on a Weyl group has a representation theoretic interpretation in terms of Verma modules. The talk concerns resulting interactions between combinatorics and homological algebra. I will present several questions around the above realization of the Bruhat order and answer them based on a series of recent works, partly joint with Volodymyr Mazorchuk and Rafael Mrden.