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

Brunat, Olivier; Neunhöffer, Max

doi:10.1090/S1088-4165-2012-00383-1

A new construction of the asymptotic algebra associated to the $q$-Schur algebra
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by Olivier Brunat and Max Neunhöffer

Represent. Theory 16 (2012), 88-107

DOI: https://doi.org/10.1090/S1088-4165-2012-00383-1

Published electronically: January 18, 2012

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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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