Enumerating conjugacy classes of graphical groups over finite fields

Let cce(G) denote the number of conjugacy classes of size e of a finite group G, and let k(G) = ∑∞ e=1 cce(G) be the class number of G. It is well known that k(GLd(Fq)) is a polynomial in q for fixed d; see [24, Ch. 1, Exercise 190]. Let Ud 6 GLd be the group scheme of upper unitriangular d×d matrices. A famous conjecture due to G. Higman [12] predicts that k(Ud(Fq)) too is given by a polynomial in q for fixed d. This has been confirmed for d 6 13 by Vera-López and Arregi [27] and for d 6 16 by Pak and Soffer [19]. The former authors also showed that the sizes of conjugacy classes of Ud(Fq) are of the form qi (see [28, §3]) and that ccqi(Ud(Fq)) is a polynomial in q − 1 with non-negative integer coefficients for i 6 d− 3 (see [26]). Many authors studied variants of Higman’s conjecture for unipotent groups derived from various types of algebraic groups; see e.g. [8].

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