Fischer-Clifford matrices of the generalized symmetric group - (a computational approach)

Abstract

Let be the cyclic group of order m and N be the direct product of n copies of . Let Sn be the symmetric group of degree n. The wreath product of with Sn is a split extension of N by Sn, called the generalized symmetric group, here denoted by B(m, n). In his Ph.D. thesis Almestady presented a combinatorial method for constructing the Fischer-Cli?ord matrices of B(m, n). However as a few examples for small values of m and n show, the manual calculation of these matrices presents formidable problems and hence a computerized approach to this combinatorial method is necessary. In a previous paper the current authors have given a computer programme that computes matrices which are row equivalent to the Fischer-Cli?ord matrices of B(2, n). Here that programme is generalized to B(m, n), where m is any positive integer. It is anticipated that with some improvements, a number of the programmes given here can be incorporated into GAP. Indeed with further development work these programmes should lead to an alternative method for computing the character table of B(m, n) in GAP.