Combinatorics related to Higman's conjecture I: Parallelogramic digraphs and dispositions

Abstract

In this paper we introduce a kind of directed graphs (digraphs) arranged in shifted rows of different lengths, which arise in a natural way related to problems of finding the number of certain families of canonical primitive connected cellular matrices of the p-Sylow Gn of GLn(q) formed by the upper unitriangular matrices over the finite field with q elements. Higman's conjecture states that the number of conjugacy classes of Gn is a polynomial in q. We associate a number, which we call the counter, to each directed graph, which gives additional information about the polynomial structure of the number of conjugacy classes. We focus on a family of digraphs, which we call parallelogramic digraphs, in which we have n rows of length k each one shifted one place to the right with respect to the previous one. We give explicit formulas for their counters for n up to 5. We prove also that the counters satisfy recurrence equations for fixed k when we vary n, proving thus a fact that was empirically observed by R.H. Harding and A.P. Heinz and proved by P. Sun for k up to 5. When n>1, this number multiplied by (q−1)nk−1 corresponds to the cardinality of the family of canonical cellular nk×nk matrices over the field Fq with n pivot lines of length k and exactly k−1 links connecting the pilots of the lines. We indicate other kinds of digraphs related to Higman's conjecture that establish lines of future research on this topic.