Representations of p′-valenced Schurian schemes

Abstract

Let p be a prime number. We consider representations of p′- valenced Schurian schemes over a field of characteristic p, especially the case that the cardinality of the underlying set can be divided by p and not by p2. A typical example of such scheme is obtained by the following way. Let G be a finite group of order pq, where q is prime to p, and let H be a p′-subgroup of G. Define the scheme by the action of G on H\G. In this case, we will show that the adjacency algebra is a direct sum of some Brauer tree algebras and simple algebras, and hence it has finite representation type. Also we give some examples of the case that G is the symmetric group of degree p and H is its Young subgroup.