Project description
Exploring character formulas for reductive algebraic groups
Funded by the European Research Council, the ModRed project will examine geometric representation theory of reductive algebraic groups over algebraically closed fields of positive characteristic. The project’s primary goal is to derive character formulas for simple and indecomposable tilting representations of these groups, by developing a geometric framework for their representation categories. Obtaining such formulas has been a significant challenge: a 1990s programme led to a formula for simple representations characters, valid only when the base field characteristic exceeds a large bound; a recent breakthrough revealed this formula does not apply to smaller characteristics. Knowledge about characters of tilting modules remains limited. ModRed will leverage new insight from the study of parity sheaves and a diagrammatic presentation of their category.
Objective
The main theme of this proposal is the Geometric Representation Theory of reductive algebraic groups over algebraically closed fields of positive characteristic. Our primary goal is to obtain character formulas for simple and for indecomposable tilting representations of such groups, by developing a geometric framework for their categories of representations.
Obtaining such formulas has been one of the main problems in this area since the 1980's. A program outlined by G. Lusztig in the 1990's has lead to a formula for the characters of simple representations in the case the characteristic of the base field is bigger than an explicit but huge bound. A recent breakthrough due to G. Williamson has shown that this formula cannot hold for smaller characteristics, however. Nothing is known about characters of tilting modules in general (except for a conjectural formula for some characters, due to Andersen). Our main tools include a new perspective on Soergel bimodules offered by the study of parity sheaves (introduced by Juteau-Mautner-Williamson) and a diagrammatic presentation of their category (due to Elias-Williamson).
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: https://op.europa.eu/en/web/eu-vocabularies/euroscivoc.
- natural sciencesmathematicspure mathematicsarithmetics
- natural sciencesmathematicspure mathematicsalgebra
- natural sciencesmathematicspure mathematicsgeometry
- natural sciencesmathematicspure mathematicsdiscrete mathematicscombinatorics
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Funding Scheme
ERC-STG - Starting GrantHost institution
63000 Clermont Ferrand
France