### Abstract

The construction by Frobenius of group characters described in Chap. 1 may be generalized to the case where a permutation group G acting on a finite set has the Gelfand pair property explained below. A general setting which encompasses and extends this is that of an association scheme. For any association scheme there is available a character theory, which in the case where the scheme arises from a group coincides with that of group characters. The development of the theory is described in this chapter. Firstly Schur investigated centralizer rings of permutation groups, then Wielandt defined S-rings over a group. The theory of association schemes provides a character theory even when a group is not present, and this can be applied to obtain a character theory for a loop or quasigroup. In particular a Frobenius reciprocity result was obtained for quasigroup characters, and it was realized that this is available for arbitrary association schemes. A further idea, that of fusion of characters of association schemes, leads to interesting results including a “magic rectangle” condition.

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

Publisher | Springer Verlag |

Pages | 137-175 |

Number of pages | 39 |

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. 137-175). (Lecture Notes in Mathematics; Vol. 2233). Springer Verlag. https://doi.org/10.1007/978-3-030-28300-1_4

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*Lecture Notes in Mathematics.*Lecture Notes in Mathematics, vol. 2233, Springer Verlag, pp. 137-175. https://doi.org/10.1007/978-3-030-28300-1_4

**S-Rings, Gelfand Pairs and Association Schemes.** / Johnson, Kenneth W.

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

TY - CHAP

T1 - S-Rings, Gelfand Pairs and Association Schemes

AU - Johnson, Kenneth W.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The construction by Frobenius of group characters described in Chap. 1 may be generalized to the case where a permutation group G acting on a finite set has the Gelfand pair property explained below. A general setting which encompasses and extends this is that of an association scheme. For any association scheme there is available a character theory, which in the case where the scheme arises from a group coincides with that of group characters. The development of the theory is described in this chapter. Firstly Schur investigated centralizer rings of permutation groups, then Wielandt defined S-rings over a group. The theory of association schemes provides a character theory even when a group is not present, and this can be applied to obtain a character theory for a loop or quasigroup. In particular a Frobenius reciprocity result was obtained for quasigroup characters, and it was realized that this is available for arbitrary association schemes. A further idea, that of fusion of characters of association schemes, leads to interesting results including a “magic rectangle” condition.

AB - The construction by Frobenius of group characters described in Chap. 1 may be generalized to the case where a permutation group G acting on a finite set has the Gelfand pair property explained below. A general setting which encompasses and extends this is that of an association scheme. For any association scheme there is available a character theory, which in the case where the scheme arises from a group coincides with that of group characters. The development of the theory is described in this chapter. Firstly Schur investigated centralizer rings of permutation groups, then Wielandt defined S-rings over a group. The theory of association schemes provides a character theory even when a group is not present, and this can be applied to obtain a character theory for a loop or quasigroup. In particular a Frobenius reciprocity result was obtained for quasigroup characters, and it was realized that this is available for arbitrary association schemes. A further idea, that of fusion of characters of association schemes, leads to interesting results including a “magic rectangle” condition.

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U2 - 10.1007/978-3-030-28300-1_4

DO - 10.1007/978-3-030-28300-1_4

M3 - Chapter

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SP - 137

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BT - Lecture Notes in Mathematics

PB - Springer Verlag

ER -