Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

On the group determinant

K. W. Johnsona1

a1 The Pennsylvania State University, Ogontz Campus, 1600 Woodland Road, Abington, PA 19001, U.S.A.

The original motivation for the introduction by Frobenius of group characters for non-abelian groups was the problem of the factorization of the group determinant corresponding to a finite group G. The original papers are [5] and [6] and a good historical survey of the work is given in [7] and [8]. If G is of order n, the group matrix XG is defined to be the n×n matrix {xg, h} where xg, h = xghxs2208G. Here the xg, gxs2208G, represent variables. The group determinant ΘG is defined to be det(XG), and is thus a polynomial of degree n in the xg. This determinant is the same, up to sign, as that of the matrix obtained from the unbordered multiplication table of G by replacing each element g by xg. If there is no ambiguity ΘG will be written as Θ.

(Received December 01 1989)

(Revised March 23 1991)