Seminar: TORSION IN GILBERT–HOWIE GROUPS
Speaker: Ihechukwu Chinyere, Department of Mathematics and Applied Mathematics, University of Pretoria, South Africa.
Abstract: At its heart, this talk is about a family of integer circulant matrices and a surprisingly subtle question about their Smith normal forms. A circulant matrix is one in which each row is a cyclic shift of the previous one; such matrices arise naturally whenever a system has discrete rotational symmetry, and their spectral theory is governed by evaluations of a single polynomial at roots of unity. The specific family we study comes from group theory: the Gilbert–Howie groups $H(n, m)$, whose relation matrices are circulants built from the polynomial $f(t) = t^m − t + 1$. These groups form a well-studied two-parameter family. Determining whether the Smith normal form of the relation matrix has no torsion part, equivalently, whether all invariant factors equal 1 so that the abelianisation $H(n, m)^{ab} is a free abelian of rank two, turns out to be a combinatorial and arithmetic problem in disguise. Noferini and Williams conjectured (2022) that $H(n, m)^{ab}\equiv\mathbb{Z}^2$ if and only if $6 | n$ and $m = 2$. We prove this conjecture. Their key observation reduces the question entirely to integer polynomials: $H(n, m)^{ab}\equiv\mathbb{Z}^2$ if and only if a certain resultant $Res(F, G)$ of two explicit polynomials equals 1. The proof that no other pair $(n, m)$ achieves this is combinatorial at its core, combining growth estimates for Lucas numbers, $p$-adic valuations of resultants, and norm bounds in cyclotomic fields via a minimal-counterexample argument. No prior knowledge of group theory is assumed.
Keywords:Circulant matrix, Smith normal forms, Gilbert–Howie groups.