The Integer group determinants for the Heisenberg group of order $p^3$

TL;DR

This paper investigates the properties of integer group determinants for the non-abelian Heisenberg group of order p^3, establishing congruences, divisibility conditions, and explicit values, with implications for Lind's Mahler measure generalizations.

Contribution

It characterizes the integer group determinants for the Heisenberg group of order p^3, including coprimality, divisibility, and explicit value conditions, and explores their relation to Lind's Mahler measure.

Findings

Established a congruence for group determinants of the Heisenberg group

Characterized all determinant values coprime to p

Determined all values for p=3 and divisibility conditions

Abstract

We establish a congruence satisfied by the integer group determinants for the non-abelian Heisenberg group of order $p^3$. We characterize all determinant values coprime to $p$, give sharp divisibility conditions for multiples of $p$, and determine all values when $p=3$. We also provide new sharp conditions on the power of $p$ dividing the group determinants for $\mathbb Z_p^2$. For a finite group, the integer group determinants can be understood as corresponding to Lind's generalization of the Mahler measure. We speculate on the Lind-Mahler measure for the discrete Heisenberg group and for two other infinite non-abelian groups arising from symmetries of the plane and 3-space.