Wreath determinants for group-subgroup pairs

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:133

Article

Hamamoto, Kei¹  Kimoto, Kazufumi²  Tachibana, Kazutoshi³  Wakayama, Masato⁴ 

[1] Kikugawa Jr High Sch, Shimonoseki, Yamaguchi 7500317, Japan

[2] Univ Ryukyus, Dept Math Sci, Nishihara, Okinawa 9030213, Japan

[3] Shuyukan High Sch, Sawara Ku, Fukuoka 8148510, Japan

[4] Kyushu Univ, Inst Math Ind, Fukuoka 8190395, Japan

关键词: Group determinants;    Wreath determinants;    Finite groups;    Symmetric groups;    Characters;    Cayley graphs;   

DOI  :  10.1016/j.jcta.2015.02.002

来源: Elsevier

【 摘 要 】

The aim of the present paper is to generalize the notion of the group determinants for finite groups. For a finite group and its subgroup H, one may define a rectangular matrix of size #H x #G by X = (x(hg)-1)h is an element of H, g is an element of G, where {xg vertical bar g is an element of G} are indeterminates indexed by the elements in G. Then, we define an invariant circle minus(G,H) for a given pair circle minus(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 km 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 circle minus(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 circle minus(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. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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