A Note on Slice Rank and Matchings in Groups
Kevin Pratt

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 is a triple of sets such that if and only if . Here we record the fact that has no multiplicative 3-matching of size greater than , yet the slice rank of its group algebra's multiplication tensor is at least over any field. This gives a negative answer to a conjecture of Petrov.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Finite Group Theory Research · Cooperative Communication and Network Coding
