S-Rings, Gelfand Pairs and Association Schemes

Research output: Chapter in Book/Report/Conference proceedingChapter

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 languageEnglish (US)
Title of host publicationLecture Notes in Mathematics
PublisherSpringer Verlag
Pages137-175
Number of pages39
DOIs
StatePublished - Jan 1 2019

Publication series

NameLecture Notes in Mathematics
Volume2233
ISSN (Print)0075-8434
ISSN (Electronic)1617-9692

Fingerprint

Gelfand Pairs
Association Scheme
Character Theory
Ring
Quasigroup
Permutation group
Frobenius
Centralizer
Reciprocity
Rectangle
Finite Set
Fusion
Character
Arbitrary

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

Johnson, K. W. (2019). S-Rings, Gelfand Pairs and Association Schemes. In 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
Johnson, Kenneth W. / S-Rings, Gelfand Pairs and Association Schemes. Lecture Notes in Mathematics. Springer Verlag, 2019. pp. 137-175 (Lecture Notes in Mathematics).
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Johnson, KW 2019, S-Rings, Gelfand Pairs and Association Schemes. in 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.

Lecture Notes in Mathematics. Springer Verlag, 2019. p. 137-175 (Lecture Notes in Mathematics; Vol. 2233).

Research output: Chapter in Book/Report/Conference proceedingChapter

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Johnson KW. S-Rings, Gelfand Pairs and Association Schemes. In Lecture Notes in Mathematics. Springer Verlag. 2019. p. 137-175. (Lecture Notes in Mathematics). https://doi.org/10.1007/978-3-030-28300-1_4