Wreath determinants for group-subgroup pairs

The aim of the present paper is to generalize the notion of the group determinants for finite groups. For a finite group G and its subgroup H, one may define a rectangular matrix of size #H×#G by X=(xhg-1)h∈H,g∈G, where {xg|g∈G} are indeterminates indexed by the elements in G. Then, we define an invariant Θ(G, H) for a given pair (G, H) by the k-wreath determinant of the matrix X, where k is the index of H in G. The k-wreath determinant of an n by kn matrix is a relative invariant of the left action by the general linear group of order n and of the right action by the wreath product of two symmetric groups of order k and n. Since the definition of Θ(G, H) is ordering-sensitive, the representation theory of symmetric groups is naturally involved. When G is abelian, if we specialize the indeterminates to powers of another variable q suitably, then Θ(G, H) factors into the product of a power of q and polynomials of the form 1-q^r for various positive integers r. We also give examples for non-abelian group-subgroup pairs.