The next meeting of the Algebra and Logic Seminar will be held on **February 17, 2023** (Friday) online in Zoom.

A talk on:

## Invariant theory for reductive subgroups of reductive groups

will be delivered by **Valdemar Tsanov (IMI-BAS)****.**

**Abstract.**Let H ⸦ G be an embedding of connected complex reductive linear algebraic groups. A classical question with several important interpretations is: which irreducible G-modules contain nonzero H-invariant vectors? An approach based on the Geometric Invariant Theory of Hilbert-Mumford was developed in works of Heckman, Berenstein-Sjamaar, Belkale-Kumar and Ressayre, culminating in a description of the generalized Littlewood-Richadson cone – the convex hull of set of the highest weights of the G-modules containing H-invariants. The discrepancy between the convex hull and the actual set of weights presents the so-called saturation problem, famously solved by Knutson and Tao for diagonal embeddings of GL_n, and widely open in general. Ressayre’s description of the cone demands extensive calculations even in relatively tame cases, which makes applications difficult. Further development of the structure theory seems desirable.

In this talk, based on joint works with Seppänen and Staneva, I will present some structural properties of generalized Littlewood-Richardson cones, allowing to partition the subgroups of a given G into types according to the properties of the cones. We derive a new numerical invariant of reductive groups, and use it to show that for “generic” subgroups the cone fills the entire Weyl chamber of G. This greatly reduces the difficulty of the saturation problem and allows it to be solved for some new cases, e.g. SL_2-subgroups of classical groups.

The seminar will be held online. Join the Zoom link:

https://us02web.zoom.us/j/

Topic: Онлайн семинар на секция “Алгебра и логика”

Time: Feb 17, 2023 01:00 PM Sofia

Meeting ID: 851 3737 5021

Passcode: 035647

Algebra and Logic Department, IMI – BAS

http://www.math.bas.bg/algebra/seminarAiL/

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