Higher Commutativity in Finite Groups, Rigidity, Extremal bounds, and Heisenberg-Type Families

TL;DR

This paper investigates the extremal and rigidity properties of the hierarchy of probabilities of commuting r-tuples in finite groups, revealing exact formulas and structural insights, especially for groups with specific algebraic properties.

Contribution

It provides new exact formulas and rigidity results for the hierarchy of commuting probabilities in finite groups, connecting representation theory and topological quantum field theory.

Findings

Derived an exact all-rank formula for cyclic-index rigidity in certain groups.

Obtained closed formulas for the class-2 exponent-p groups and the $ ext{Heisenberg}$ family.

Proved that the pair $(P_2(G),P_3(G))$ determines the isoclinism class within the $ ext{Heisenberg}$ family.

Abstract

For a finite group $G$ and an integer $r\ge 2$ let $$ P_r(G):=\frac{|Hom(\mathbb Z^r,G)|}{|G|^r}, $$ where $\Hom(\mathbb Z^r,G)$ is the set of pairwise commuting $r$-tuples in $G$. This paper studies rigidity and extremal behavior of the hierarchy $\{P_r(G)\}_{r\ge2}$, together with a low-rank representation-theoretic / TQFT counting bridge. The first main direction is cyclic-index rigidity: for groups with an abelian normal subgroup $A$ and cyclic quotient of order $\omega$, under a natural fixed-subgroup hypothesis we prove the exact all-rank formula $$ P_r(G)=\frac{1}{\omega^r}+\left(1-\frac{1}{\omega^r}\right)\left(\frac{|A\cap Z(G)|}{|A|}\right)^{r-1}, $$ which yields gap and rigidity statements for non-abelian abelian extensions of prime index. The second main direction is the class-$2$ exponent-$p$ world. We develop a symplectic reduction, obtain closed formulas when $|G'|=p$, and prove a closed all-$r$ hierarchy in the $\mathbb F_q$-Heisenberg family: \[ P_r(G)=q^{-2nr}\sum_{k=0}^{\min(n,r)}L_{n,k}(q)\prod_{i=0}^{k-1}(q^r-q^i). \] In particular, inside the $\mathbb F_q$-Heisenberg family the pair $(P_2(G),P_3(G))$ already determines the isoclinism class. Combining the cyclic-index formula with the known sharp upper bound for the multiple commutativity degree gives equality and near-extremal rigidity, including a stability gap near $11/32$ for commuting triples. At the low-rank end we also prove explicit class-number formulas for $P_3(G)$ and $P_4(G)$; these recover the simple-count formulas for the untwisted Drinfeld double and the untwisted quantum triple / double-loop-groupoid algebra.