Commutative Schur Rings of Maximal Dimension |
| |
| Authors: | Stephen P Humphries Kenneth W Johnson Andrew Misseldine |
| |
| Institution: | 1. Department of Mathematics , Brigham Young University , Provo , Utah , USA steve@mathematics.byu.edu;3. Abington College , Pennsylvania State University, Ogontz Campus , Abington , Pennsylvania , USA;4. Department of Mathematics , Brigham Young University , Provo , Utah , USA |
| |
| 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. |
| |
| Keywords: | Finite group Frobenius group Group matrix Metacyclic group Random walk S-ring Special linear group |
|
|
