Representations of p′-valenced Schurian schemes

Abstract

Let $p$ be a prime number. We consider representations of $p\prime$- 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 $p^2$. 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 $p\prime$-subgroup of $G$. Define the scheme by the action of $G$ on $H\backslash G$. 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.