A Note on Slice Rank and Matchings in Groups

A multiplicative 3-matching in a group $G$ is a triple of sets $\{a_i\},\{b_i\},\{c_i\}\subset G$ such that $a_i{b}_j{c}_k=1$ 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 $O(p^{8\mathrm{/}3})$, yet the slice rank of its group algebra’s multiplication tensor is at least $\mathrm{\Omega }(p^3)$ over any field. This gives a negative answer to a conjecture of Petrov.