Representation type of \(q\)-Schur algebras (Q2750920)

Classical and quantised Schur algebras are crucial for representation theory and cohomology of finite and infinite general linear groups. One of the basic questions on Schur algebras is a classification problem, to decide when a Schur algebra is semisimple, or of finite, of tame or of wild representation type. Earlier work by the authors, joint with various coauthors [see Q. J. Math., II. Ser. 44, No. 173, 17-41 (1993; Zbl 0832.16011), Math. Proc. Camb. Philos. Soc. 124, No. 1, 15-20 (1998; Zbl 0916.20031), Math. Z. 232, No. 1, 137-182 (1999; Zbl 0942.16021)] has completely solved this problem for classical Schur algebras. The present article gives a complete answer for quantised Schur algebras. The answer depends on the three parameters \(n\) (the size of the general linear group), \(r\) (the degree) and \(q\). Semisimplicity occurs if \(q\) is not a root of unity of small order, and also for some \(n=2\) cases. Other finite type cases occur for \(n=2\) and for roots of unity of bigger order. There are tame cases for \(n=2,3,4\). The article combines general reduction methods with very explicit information about certain Schur algebras. The results demonstrate the high quality of the present knowledge on Schur algebras. Much less is known, however, about blocks of Schur algebras; for blocks the classification problem also is of interest (and wide open).