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

Abstract

Let Zm be the cyclic group of order m and N be the direct product of n
copies of Zm. Let S_n be the symmetric group of degree n. The wreath product of
Z_m with S_n is a split extension of N by S_n, 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-Clifford 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-Clifford 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.

Mathematics Subject Classication (2010): 20C15, 20C30, 20C40, 05E15, 05E18, 20E22.

Key words: Generalized symmetric group, m-compositions of n, Fischer-Clifford matrices,
m-set of partition [λ].