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Representation Theory

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A new construction of the asymptotic algebra associated to the $q$-Schur algebra

Authors: Olivier Brunat and Max Neunhöffer
Journal: Represent. Theory 16 (2012), 88-107
MSC (2010): Primary 20C08, 20F55; Secondary 20G05
Published electronically: January 18, 2012
MathSciNet review: 2869019
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Abstract: We denote by $A$ the ring of Laurent polynomials in the indeterminate $v$ and by $K$ its field of fractions. In this paper, we are interested in representation theory of the “generic” $q$-Schur algebra $\mathcal {S}_q(n,r)$ over $A$. We will associate to every symmetrising trace form $\tau$ on $K\mathcal {S}_q(n,r)$ a subalgebra $\mathcal {J}_{\tau }$ of $K\mathcal {S}_q(n,r)$ which is isomorphic to the “asymptotic” algebra $\mathcal {J}(n,r)_A$ defined by J. Du. As a consequence, we give a new hypothesis which implies James’ conjecture.

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Additional Information

Olivier Brunat
Affiliation: Ruhr-Universität Bochum, Fakultät für Mathematik, D-44780 Bochum, Germany
Address at time of publication: Institut de Mathèmatiques de Jussieu, UFR de Mathèmatiques, 175, rue du Chevaleret, F-75013 Paris

Max Neunhöffer
Affiliation: School of Mathematics and Statistics, Mathematical Institute, North Haugh, St Andrews, Fife KY16 9SS, Scotland, United Kingdom

Received by editor(s): January 9, 2009
Received by editor(s) in revised form: April 2, 2010
Published electronically: January 18, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.