Schur extensions of monomial matrix groups

Abstract

For each Schur function, or generalized matrix function, corresponding to a linear character of a subgroup of the symmetric group, we construct a matrix semigroup containing the monomial group defined by the subgroup. The matrix semigroup constructed has the property that the generalized matrix function is multiplicative on it and it is maximal with respect to the inclusion of transvections. It is further shown, with mild restrictions on the underlying field, that every matrix semigroup with the above properties is obtained in this way.