Commutative Schur Rings of Maximal Dimension

Abstract

A commutative Schur ring over a finite group G has dimension at most s _G  = d ₁ + … +d _r , where the d _i 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, 2 ⁿ ), 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.

Key Words:

• Finite group
• Frobenius group
• Group matrix
• Metacyclic group
• Random walk
• S-ring
• Special linear group

2010 Mathematics Subject Classification:

• 20C05
• 05E30

ACKNOWLEDGMENT

All calculations made in the preparation of this paper were accomplished using Magma [18]. We are grateful to a referee for his/her comments.

Notes

Communicated by A. Turull.