Calculating conjugacy classes in Sylow p-subgroups of finite Chevalley groups of rank six and seven

TL;DR

This paper extends computational methods to determine the number of conjugacy classes in Sylow p-subgroups of finite Chevalley groups of ranks six and seven, confirming the polynomial nature predicted by a generalized Higman conjecture.

Contribution

It refines and improves an algorithm to compute conjugacy class counts for higher-rank Chevalley groups, validating the polynomial conjecture for these cases.

Findings

k(U(q)) is a polynomial in q for ranks six and seven

Coefficients of k(U(q)) in q-1 are non-negative

Explicit formulas provided for coefficients of degrees zero, one, and two

Abstract

Let G(q) be a finite Chevalley group, where q is a power of a good prime p, and let U(q) be a Sylow p-subgroup of G(q). Then a generalized version of a conjecture of Higman asserts that the number k(U(q)) of conjugacy classes in U(q) is given by a polynomial in q with integer coefficients. In an earlier paper, the first and the third authors developed an algorithm to calculate the values of k(U(q)). By implementing it into a computer program using GAP, they were able to calculate k(U(q)) for G of rank at most 5, thereby proving that for these cases k(U(q)) is given by a polynomial in q. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of k(U(q)) for finite Chevalley groups of rank six and seven, except E_7. We observe that k(U(q)) is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write k(U(q)) as a polynomial in q-1, then the coefficients are non-negative. Under the assumption that k(U(q)) is a polynomial in q-1, we also give an explicit formula for the coefficients of k(U(q)) of degrees zero, one and two.