133 research outputs found

    Wreath determinants for group-subgroup pairs

    Full text link
    The aim of the present paper is to generalize the notion of the group determinants for finite groups. For a finite group GG of order knkn and its subgroup HH of order nn, one may define an nn by knkn matrix X=(xhgβˆ’1)h∈H,g∈GX=(x_{hg^{-1}})_{h\in H,g\in G}, where xgx_g (g∈Gg\in G) are indeterminates indexed by the elements in GG. Then, we define an invariant Θ(G,H)\Theta(G,H) for a given pair (G,H)(G,H) by the kk-wreath determinant of the matrix XX, where kk is the index of HH in GG. The kk-wreath determinant of nn by knkn matrix is a relative invariant of the left action by the general linear group of order kk and right action by the wreath product of two symmetric groups of order kk and nn. Since the definition of Θ(G,H)\Theta(G,H) is ordering-sensitive, representation theory of symmetric groups are naturally involved. In this paper, we treat abelian groups with a special choice of indeterminates and give various examples of non-abelian group-subgroup pairs.Comment: 12 pages, 2 figure
    • …
    corecore