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

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 $\{P_r(G){\}}_{r\ge 2}$, 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 $|G^{\prime }|=p$, 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.