Tag - Schubert varieties

Thomas Haines: Pavings of convolution fibres and applications

1st March, 2024

A convolution morphism is the geometric analogue of the convolution of functions in a Hecke algebra. The properties of fibres of convolution morphisms are used in a variety of ways in the geometric Langlands programme and in the study of Schubert varieties. I will explain a very general result about cellular pavings of fibres of convolution morphisms in the setting of partial affine flag varieties, as well as applications related to the very purity and parity vanishing of cohomology of Schubert varieties over finite fields, structure constants for parahoric Hecke algebras, and the (motivic) geometric Satake equivalence.

Ed Richmond: The isomorphism problem for Schubert varieties

28th March, 2022

Schubert varieties in the full flag variety of Kac-Moody type are indexed by elements of the corresponding Weyl group. In this talk, I will discuss recent work with William Slofstra where we give a practical criterion for when two such Schubert varieties (from potentially different flag varieties) are isomorphic, in terms of the Cartan matrix and reduced words for the indexing Weyl group elements. As a corollary, we show that two such Schubert varieties are isomorphic if and only if there is an isomorphism between their integral cohomology rings that preserves the Schubert basis. As an application, we show that the isomorphism classes of Schubert varieties in a given flag variety are controlled by graph automorphisms of the Dynkin diagram.

Marc Besson: Twisted affine Schubert varieties and twisted affine Demazure modules

24th January, 2022

In this talk I will discuss work (joint with J. Hong) on the T-fixed subscheme of twisted affine Schubert varieties. This may be viewed as an extension of results of Zhu in the untwisted case. I will discuss the Frenkel-Kac-Segal isomorphism, as well as Zhu's geometrization of this result. An application of this work is the determination of the smooth locus of twisted affine Schubert varieties.

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.

William Graham: Tangent spaces and T-invariant curves of Schubert varieties

20th September, 2021

The set of weights of T-invariant curves to a Schubert variety at a T-fixed point admits a simple description due to Carrell and Peterson. The set of weights to the tangent space is much more complicated to describe, and has only been explicitly described in classical types. The main result is that although these two sets of weights are different, they generate the same cone in the dual of the Lie algebra of T.

Geordie Williamson: Spectra in representation theory

18th June, 2021

In geometric representation theory cohomology, intersection cohomology and constructible sheaves show up everywhere. This might seem strange to an algebraic topologist, who might ask: why this emphasis on cohomology, when there are so many other interesting cohomology theories (like K-theory, elliptic cohomology, complex cobordism, ...) out there? They might also ask: is there something like "intersection K-theory", or "intersection complex cobordism"? This is something I've often wondered about. I will describe work in progress with Ben Elias, where we use Soergel bimodules to investigate what KU-modules look like on the affine Grassmannian. We have checked by hand that in types A1, A2 and B2, one gets something roughly resembling the quantum group. Speaking very roughly, the intersection K-theory of Schubert varieties in the affine Grassmannian should recover the irreducible representations of the quantum group. Inspirations for this work include a strange Cartan matrix discovered by Ben Elias, and work of Cautis-Kamnitzer.

William Graham: Tangent spaces and T-stable curves in Schubert varieties

22nd March, 2021

The set of T-stable curves in a Schubert variety through a T-fixed point is relatively easy to understand, but the tangent space is more difficult. In this talk we describe some new relations between the tangent space and the set of T-fixed curves. This is joint work with Victor Kreiman.

Mahir Can: Spherical Varieties and Combinatorics

29th September, 2020

Let G be a reductive complex algebraic group with a Borel subgroup B. A spherical G-variety is an irreducible normal G-variety X where B has an open orbit. If X is affine, or if it is projective but endowed with a G-linearized ample line bundle, then the group action criteria for the sphericality is in fact equivalent to the representation theoretic statement that a certain space of functions (related to X) is multiplicity-free as a G-module. In this talk, we will discuss the following question about a class of spherical varieties: if X is a Schubert variety for G, then when do we know that X is a spherical L-variety, where L is the stabilizer of X in G.