Commutative Schur Rings of Maximal Dimension

Stephen P. Humphries, Kenneth Johnson, Andrew Misseldine

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

A commutative Schur ring over a finite group G has dimension at most s G = d1 + … +dr, where the di are the degrees of the irreducible characters of G. We find families of groups that have S-rings that realize this bound, including the groups SL(2, 2n), metacyclic groups, extraspecial groups, and groups all of whose character degrees are 1 or a fixed prime. We also give families of groups that do not realize this bound. We show that the class of groups that have S-rings that realize this bound is invariant under taking quotients. We also show how such S-rings determine a random walk on the group and how the generating function for such a random walk can be calculated using the group determinant.

Original languageEnglish (US)
Pages (from-to)5298-5327
Number of pages30
JournalCommunications in Algebra
Volume43
Issue number12
DOIs
StatePublished - Dec 2 2015

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Schur Ring
Commutative Ring
Ring
Random walk
Metacyclic Group
Character Degrees
Irreducible Character
Generating Function
Determinant
Quotient
Finite Group

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

Humphries, Stephen P. ; Johnson, Kenneth ; Misseldine, Andrew. / Commutative Schur Rings of Maximal Dimension. In: Communications in Algebra. 2015 ; Vol. 43, No. 12. pp. 5298-5327.
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Commutative Schur Rings of Maximal Dimension. / Humphries, Stephen P.; Johnson, Kenneth; Misseldine, Andrew.

In: Communications in Algebra, Vol. 43, No. 12, 02.12.2015, p. 5298-5327.

Research output: Contribution to journalArticle

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