A Note on Slice Rank and Matchings in Groups

TL;DR

This paper investigates the relationship between slice rank and multiplicative matchings in groups, showing that certain groups have high slice rank but limited matchings, thus disproving a previous conjecture.

Contribution

It demonstrates that the group PSL(2,p) has a high slice rank yet limited multiplicative 3-matchings, providing a counterexample to Petrov's conjecture.

Findings

PSL(2,p) has no large multiplicative 3-matching beyond O(p^{8/3})

The slice rank of the group algebra's multiplication tensor is at least Ω(p^3)

Disproves Petrov's conjecture relating matchings and slice rank

Abstract

A multiplicative 3-matching in a group $G$ is a triple of sets $\{a_i\}, \{b_i\}, \{c_i\} \subset G$ such that $a_ib_jc_k = 1$ if and only if $i=j=k$. Here we record the fact that $\text{PSL}(2,p)$ has no multiplicative 3-matching of size greater than $O(p^{8/3})$, yet the slice rank of its group algebra's multiplication tensor is at least $\Omega(p^3)$ over any field. This gives a negative answer to a conjecture of Petrov.