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| Formato: | Preprint |
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2026
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| Acceso en línea: | https://arxiv.org/abs/2605.17171 |
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| _version_ | 1866913136216375296 |
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| author | Levit, Vadim E Shwartz, Robert |
| author_facet | Levit, Vadim E Shwartz, Robert |
| contents | 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 $ω$, under a natural fixed-subgroup hypothesis we prove the exact all-rank formula $$ P_r(G)=\frac{1}{ω^r}+\left(1-\frac{1}{ω^r}\right)\left(\frac{|A\cap Z(G)|}{|A|}\right)^{r-1}, $$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17171 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Higher Commutativity in Finite Groups, Rigidity, Extremal bounds, and Heisenberg-Type Families Levit, Vadim E Shwartz, Robert Group Theory 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 $ω$, under a natural fixed-subgroup hypothesis we prove the exact all-rank formula $$ P_r(G)=\frac{1}{ω^r}+\left(1-\frac{1}{ω^r}\right)\left(\frac{|A\cap Z(G)|}{|A|}\right)^{r-1}, $$ 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. |
| title | Higher Commutativity in Finite Groups, Rigidity, Extremal bounds, and Heisenberg-Type Families |
| topic | Group Theory |
| url | https://arxiv.org/abs/2605.17171 |