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

TL;DR

This paper introduces a new construction of the asymptotic algebra linked to the q-Schur algebra, providing insights into its representation theory and a new criterion related to James' conjecture.

Contribution

It presents a novel method to construct the asymptotic algebra associated with q-Schur algebras using symmetrising trace forms.

Findings

Associates a subalgebra J_{ au} to each non-degenerate symmetrising trace form.

Shows J_{ au} is isomorphic to the asymptotic algebra \\J(n,r)_A.

Provides a new criterion for James' conjecture.

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 S_q(n,r) over A. We will associate to every non-degenerate symmetrising trace form \tau on KS_q(n,r) a subalgebra J_{\tau} of KS_q(n,r) which is isomorphic to the "asymptotic" algebra \J(n,r)_A defined by J. Du. As a consequence, we give a new criterion for James' conjecture.