On representation types of \(q\)-Schur algebras
The main results of this paper concern the representation type of Schur algebras and their generalizations called \(q\)-Schur algebras. Let \(S_ k(n,r)\) be a Schur algebra, where \(k\) is an algebraically closed field of prime characteristic \(p>0\) and \(n\geq r\). The first result states that \(S_ k(n,r)\) is representation-finite (that is, there are only finitely many indecomposable non-isomorphic modules over \(S_ k(n,r)\)) if and only if \(r<2p\) and, in this case, the relations for \(S_ k(n,r)\) are generated by elements of degree 2 (i.e., the algebra is quadratic). The representation-infinite algebras can be divided into two classes (tame or wild) according to the hardness of the study of their category of finitely generated modules. The author shows in this paper that representation-infinite Schur algebras \(S_ k(n,r)\), \(r\leq n\), are always wild. Now let \(S_ k(n,r,q)\), \(r\leq n\), be a \(q\)-Schur algebra, where \(k\) is an algebraically closed field of arbitrary characteristic and let \(m\) be the minimal number such that \(1+q+\cdots+ q^{m-1}\) is equal to zero. The author has shown that if \(r<2m\), then \(S_ k(n,r,q)\) is representation-finite and quadratic.
scientific article; zbMATH DE number 166164