### Abstract

The construction and properties of group matrices are analyzed in more detail. The group matrix of a cyclic group with a certain ordering is a circulant matrix. The book by Davis (Circulant Matrices, Chelsea, New York, 1994) gives a comprehensive account of circulants and the chapter is designed to provide a far reaching extension and generalization of the results there. If an arbitrary subgroup H of a group G is taken, it is shown that with an appropriate ordering the group matrix X_{G} is a block matrix in which each block is of the form where the elements in are all distinct and of the form This leads to methods to partially diagonalize the group matrix which facilitate the calculation of the group determinant. Group matrix versions of the ring of representations and the Burnside ring of a group are described using supermatrices. A description of projective group matrices, corresponding to projective representations, is given. Work of L. E. Dickson on group matrices and group determinants over a finite field is also described.

Original language | English (US) |
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Title of host publication | Lecture Notes in Mathematics |

Publisher | Springer Verlag |

Pages | 55-110 |

Number of pages | 56 |

DOIs | |

State | Published - Jan 1 2019 |

### Publication series

Name | Lecture Notes in Mathematics |
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Volume | 2233 |

ISSN (Print) | 0075-8434 |

ISSN (Electronic) | 1617-9692 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Lecture Notes in Mathematics*(pp. 55-110). (Lecture Notes in Mathematics; Vol. 2233). Springer Verlag. https://doi.org/10.1007/978-3-030-28300-1_2

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*Lecture Notes in Mathematics.*Lecture Notes in Mathematics, vol. 2233, Springer Verlag, pp. 55-110. https://doi.org/10.1007/978-3-030-28300-1_2

**Further Group Matrices and Group Determinants.** / Johnson, Kenneth W.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Further Group Matrices and Group Determinants

AU - Johnson, Kenneth W.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The construction and properties of group matrices are analyzed in more detail. The group matrix of a cyclic group with a certain ordering is a circulant matrix. The book by Davis (Circulant Matrices, Chelsea, New York, 1994) gives a comprehensive account of circulants and the chapter is designed to provide a far reaching extension and generalization of the results there. If an arbitrary subgroup H of a group G is taken, it is shown that with an appropriate ordering the group matrix XG is a block matrix in which each block is of the form where the elements in are all distinct and of the form This leads to methods to partially diagonalize the group matrix which facilitate the calculation of the group determinant. Group matrix versions of the ring of representations and the Burnside ring of a group are described using supermatrices. A description of projective group matrices, corresponding to projective representations, is given. Work of L. E. Dickson on group matrices and group determinants over a finite field is also described.

AB - The construction and properties of group matrices are analyzed in more detail. The group matrix of a cyclic group with a certain ordering is a circulant matrix. The book by Davis (Circulant Matrices, Chelsea, New York, 1994) gives a comprehensive account of circulants and the chapter is designed to provide a far reaching extension and generalization of the results there. If an arbitrary subgroup H of a group G is taken, it is shown that with an appropriate ordering the group matrix XG is a block matrix in which each block is of the form where the elements in are all distinct and of the form This leads to methods to partially diagonalize the group matrix which facilitate the calculation of the group determinant. Group matrix versions of the ring of representations and the Burnside ring of a group are described using supermatrices. A description of projective group matrices, corresponding to projective representations, is given. Work of L. E. Dickson on group matrices and group determinants over a finite field is also described.

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U2 - 10.1007/978-3-030-28300-1_2

DO - 10.1007/978-3-030-28300-1_2

M3 - Chapter

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T3 - Lecture Notes in Mathematics

SP - 55

EP - 110

BT - Lecture Notes in Mathematics

PB - Springer Verlag

ER -