Complexity and randomness in the Heisenberg groups (and beyond)

TL;DR

This paper explores the pseudo-random and random behaviors in the commuting graphs of conjugacy classes of Heisenberg groups and their limits, providing insights into the complexity of character theory in related algebraic structures.

Contribution

It analyzes the transition from finite to infinite groups, revealing how randomness manifests in algebraic and graph-theoretic contexts, and relates this to the complexity of character theory.

Findings

Pseudo-random behavior in finite Heisenberg groups

Emergence of random graph structure in the limit

Insights into the complexity of character theory

Abstract

By studying the commuting graphs of conjugacy classes of the sequence of Heisenberg groups $H_{2n+1}(p)$ and their limit $H_\infty(p)$ we find pseudo-random behavior (and the random graph in the limiting case). This makes a nice case study for transfer of information between finite and infinite objects. Some of this behavior transfers to the problem of understanding what makes understanding the character theory of the uni-upper-triangular group (mod p) "wild". Our investigations in this paper may be seen as a meditation on the question: is randomness simple or is it complicated?