2014/06/10 by Kei Hamamoto, Hamamoto, Kei, Kazufumi Kimoto +5
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #math.CO
paper · pdf · doi:10.48550/arxiv.1406.2425
12 pages, 2 figures
arxiv created 2014/06/10 · openalex publication_date 2014/06/10 · arxiv updated 2014/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The aim of the present paper is to generalize the notion of the group determinants for finite groups. For a finite group G of order kn and its subgroup H of order n, one may define an n by kn matrix 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 n by kn matrix is a relative invariant of the left action by the general linear group of order k and 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, 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.