A Generalization of Redfield's Master Theorem

Abstract

Generalizations of Redfield's master theorem and superposition theorem are proved by using decomposition of the tensor product of several induced monomial representations of the symmetric group $S_d$ into transitive constituents. As direct consequences, one obtains several graphical corollaries. Given graphs $Γ_1,\hdots ,Γ_k$, with $d$ vertices, together with their automorphism groups $W_1\leq S_d,\hdots, W_k\leq S_d$, one can find the number of superpositions of $Γ_1,\hdots ,Γ_k$, whose automorphism groups satisfy one of the following conditions: (1) the groups consist of even permutations; (2) the groups are trivial, in case at least one of $W_m$'s is cyclic; (3) the groups are of odd order, in case at least one of $W_m$'s is dihedral and its order is not divisible by 4; (4) the groups are of order dividing a natural number $r$, in case at least one of $W_m$'s has a normal solvable subgroup of order $r$, such that the corresponding factor-group is cyclic of order relatively prime to $r$; (5) the groups are $q$-groups ($q$ is a prime), in case at least one of $W_m$'s has a normal $q$-subgroup such that the corresponding factor-group is cyclic of order relatively prime to $q$.