### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 76-96 |

Number of pages | 21 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 133 |

DOIs | |

Publication status | Published - Jul 1 2015 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Combinatorial Theory. Series A*,

*133*, 76-96. https://doi.org/10.1016/j.jcta.2015.02.002

**Wreath determinants for group-subgroup pairs.** / Hamamoto, Kei; Kimoto, Kazufumi; Tachibana, Kazutoshi; Wakayama, Masato.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory. Series A*, vol. 133, pp. 76-96. https://doi.org/10.1016/j.jcta.2015.02.002

}

TY - JOUR

T1 - Wreath determinants for group-subgroup pairs

AU - Hamamoto, Kei

AU - Kimoto, Kazufumi

AU - Tachibana, Kazutoshi

AU - Wakayama, Masato

PY - 2015/7/1

Y1 - 2015/7/1

N2 - 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-qr for various positive integers r. We also give examples for non-abelian group-subgroup pairs.

AB - 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-qr for various positive integers r. We also give examples for non-abelian group-subgroup pairs.

UR - http://www.scopus.com/inward/record.url?scp=84923026611&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923026611&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2015.02.002

DO - 10.1016/j.jcta.2015.02.002

M3 - Article

AN - SCOPUS:84923026611

VL - 133

SP - 76

EP - 96

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

ER -