Fischer matrices of the affine groups (Q5943069)

Let \(p\) be a prime and let \(q=p^a\), where \(a\) is a positive integer. Let \(\text{GL}_n(q)\) denote the general linear group of degree \(n\) over the field of \(q\) elements and let \(V_n(q)\) denote the natural vector space of dimension \(n\) on which \(\text{GL}_n(q)\) acts. The affine group \(H_n\) is the semi-direct product \(V_n(q)\cdot\text{GL}_n(q)\). Writing \(H\), \(V\) and \(G\) in place of \(H_n\), \(V_n(q)\) and \(\text{GL}_n(q)\), respectively, the problem addressed in the paper is to calculate the irreducible complex characters of \(H\). Since \(H\) is a semi-direct product with Abelian normal subgroup \(V\), elementary Clifford theory applies. Indeed, \(G\) has a single orbit on the non-principal irreducible characters of \(V\) and the stabilizer subgroup of any such character is isomorphic to \(H_{n-1}\). Thus it seems that we have a good inductive situation for trying to calculate characters of \(H_n\). As far as we can tell, Fischer matrices provide a means of encoding data relevant to this situation, but the paper does not provide many details on exactly how they are used. Suffice it to say that the Fischer matrices produced have integer entries, many having the form \(q^a-q^b\), where \(a\) and \(b\) are non-negative integers with \(a\geq b\).