CAT REPTH · Categorification in Representation Theory
FP7 — People (Marie Curie Actions)
- Duration
- 2010-09-01 → 2013-08-31
- EU contribution
- €45,000
- Participants
- 1
- Scheme
- MC-ERG
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Results in brief
Categorification in Representation Theory
Executive Summary: Representation theory is a part of pure mathematics which aims to understand structures arising from symmetries on certain spaces. It has long been known that many two-dimensional geometric phenomena have algebraic explanations via the subject of representation theory. In the last fifteen years it has been conjectured that analogous explanations for problems in three-dimensional geometry should raise from introducing so-called higher symmetries, giving rise to the subject of 2-representation theory, or categorification. The project has investigated to instances of higher representation theory. The first, building on the fellow's expertise acquired during her Marie-Curie Intra-European Fellowship, investigated, in collaboration with Will Turner (University of Aberdeen), the use of higher representation theoretic methods in understanding all symmetries of the plane. In particular it gave detailed homological information in terms of computing invariants of of these symmetries. The second instance built on the fellow's research experience prior to her Marie-Curie Intra-European Fellowship, and investigated certain prominent algebras in the theory of categorification, the so-called Khovanov-Lauda-Rouquier algebras which have close connections to affine Hecke algebras. In collaboration with Alexander Kleshchev and Joseph Loubert (both University of Oregon) respectively Jérémie Guilhot (then University of East Anglia, now Université de Tours), affine cellularity of Khovanov-Lauda-Rouquier algebras in finite type A respectively of affine Hecke algebras of rank two was proved, thus giving important information about their homological structure. Furthermore the fellow, in joint work with Volodymyr Mazorchuk (University of Uppsala), extended the notion of cellularity to abstract 2-representations of fiat 2-categories, thus initiating a general study of 2-represenations of 2-categories, which had up to date only been studied in examples. Contact details: Dr Vanessa Miemietz School of Mathematics University of East Anglia Norwich NR4 7TJ email: v.miemietz@uea.ac.uk Website: http://www.uea.ac.uk/~byr09xgu/
Data: CORDIS, © European Union
Project objective
The subject of representation theory originally arose from the study of symmetries on certain spaces. It has long been known that many two-dimensional geometric phenomena have algebraic explanations via the subject of representation theory. In the last fifteen years it has been conjectured that analogous explanations for problems ins three-dimensional geometry should raise from introducing so-called higher symmetries, giving rise to the subject of 2-representation theory, or categorification. This is the newly developing research area where this proposal is situated.The proposal has two main objectives:In her previous Marie Curie fellowship the applicant and her co-author have made significant progress on the representation theory of general linear groups over a field of positive characteristic by developing higher categorical methods to iteratively generate categories of representation for the general linear group of rank two. This yields very strong results on the categories of representations in this case, such as e.g. multi-gradings and braid group actions on their derived categories. The first aim of this proposal is to generalise these concepts to general linear groups of larger rank.The second objective investigates affine Hecke algebras, the applicant's main area of expertise prior to her Marie Curie fellowship, in the context of categorification. It applies newly developed concepts to algebras defined by Khovanov-Lauda-Rouquier and Varagnolo-Vasserot, which are known to have very similar representation theory to affine Hecke algebras of different types. The goal of this part of the project is to gain a thorough understanding of the homological structures of these algebras.
Original text from CORDIS.
Participants
- UNIVERSITY OF EAST ANGLIA · NorwichCoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union