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MODULAR DECOMPOSITION NUMBERS OF CYCLOTOMIC HECKE AND DIAGRAMMATIC CHEREDNIK ALGEBRAS: A PATH THEORETIC APPROACH

Part of: Representation theory of groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  03 July 2018

C. BOWMAN  and
A. G. COX
C. BOWMAN
Affiliation:
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK; [email protected]
A. G. COX
Affiliation:
Department of Mathematics, City, University of London, London, UK; [email protected]

Abstract

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We introduce a path theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher-level generalizations over fields of arbitrary characteristic. Our first main result is a ‘super-strong linkage principle’ which provides degree-wise upper bounds for graded decomposition numbers (this is new even in the case of symmetric groups). Next, we generalize the notion of homomorphisms between Weyl/Specht modules which are ‘generically’ placed (within the associated alcove geometries) to cyclotomic Hecke and diagrammatic Cherednik algebras. Finally, we provide evidence for a higher-level analogue of the classical Lusztig conjecture over fields of sufficiently large characteristic.

MSC classification

Primary: 20C08: Hecke algebras and their representations
Secondary: 20G43: Schur and $q$-Schur algebras
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 6 , 2018 , e11
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2018

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