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STABILITY PATTERNS IN REPRESENTATION THEORY

Part of: Linear algebraic groups and related topics Algebraic combinatorics Categories with structure General commutative ring theory

Published online by Cambridge University Press:  15 June 2015

STEVEN V SAM  and
ANDREW SNOWDEN
STEVEN V SAM
Affiliation:
Department of Mathematics, University of California, Berkeley, CA, USA; svs@math.berkeley.edu
ANDREW SNOWDEN
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI, USA; asnowden@umich.edu

Abstract

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We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results: for example, the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, and a canonical derived auto-equivalence of the general linear theory.

MSC classification

Primary: 05E05: Symmetric functions and generalizations 20G05: Representation theory
Secondary: 13A50: Actions of groups on commutative rings; invariant theory 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , 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/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

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