Let $G$ be a finite group and $S$ be a finite set, with $G$ acting on $S$. I consider indeterminates $x_g$ indexed by $g\in G$ and form the matrix of the group action $A\in M_{S\times S}$. Its entries are indexed by pairs $(s,t)$ of elements of $S$. By definition, $a_{st}$ is the sum of the $x_g$'s over the group elements such that $s^g=t$.

What are the irreducible factors of $\det A$ ?

Without loss of generality, we may assume that the action is transitive, because otherwise $A$ is block diagonal. Just sort out the elements of $S$ by orbits. We may also assume that the action is faithful.

This determinant has an obvious factor $\ell:=\sum_{g\in G}x_g$, because $\ell$ is the sum of the entries of every row (or of every column). What does it mean about the action that the quotient $(\det A)/\ell$ be irreducible ?

*Notice* that if $S=G$ and $G$ acts by multipication, this determinant is that considered by Dedekind and Frobenius, which led the latter to the theory of representations.

free. – Tom Church May 4 '11 at 5:03