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-qr 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 |
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All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
Cite this
Wreath determinants for group-subgroup pairs. / Hamamoto, Kei; Kimoto, Kazufumi; Tachibana, Kazutoshi; Wakayama, Masato.
In: Journal of Combinatorial Theory. Series A, Vol. 133, 01.07.2015, p. 76-96.Research output: Contribution to journal › Article
}
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.
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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 -