Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5772984 | Linear Algebra and its Applications | 2017 | 27 Pages |
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 (qâ1)nkâ1 corresponds to the cardinality of the family of canonical cellular nkÃnk matrices over the field Fq with n pivot lines of length k and exactly kâ1 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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Antonio Vera López, Luis MartÃnez, Antonio Vera Pérez, Beatriz Vera Pérez, Olga Basova,