\(q\)-Schur algebras, asymptotic forms, and quantum \(SL_ n\) (Q1903022)

An important application of Kazhdan-Lusztig bases and positivity properties for Hecke algebras are the asymptotic methods introduced by Lusztig in the study of cell classification of affine Weyl groups and representations of Hecke algebras. Following the author's previous work, this paper is an attempt to investigate asymptotic methods for \(q\)-Schur algebras and related quantum groups. He introduces some \(\mathbb{Z}\)-free algebras \({\mathcal T} (n, r)\) of finite rank, called the asymptotic forms of \(q\)-Schur algebras, and studies their structure and representations. Also he shows how these algebras are linked with a certain asymptotic form for the quantum special linear group and how they behave in forming the completion of the quantized enveloping algebras for \(GL_n\) via \(q\)-Schur algebras.