Time: 9h30, Thursday, May 7, 2020.

Location: Room 612, Building A6, Institutte of Mathematics.

Abstract: We introduce the notion of symmetric subrank of tensors. The symmetric subrank of a tensor $t in (mathbb{F}^d)^{otimes k}$ is defined as the largest integer $r$ such that the unit tensor $left langle r right rangle$ can be written as $left langle r right rangle = A^{otimes k} t$ for some matrix $r times d$ matrix $A$. We prove some basic properties for the symmetric subrank and give examples where it is different from the subrank. We give an upper bound on the asymptotic symmetric subrank.

Our main motivation comes from the problem of computing the Shannon capacity of hypergraphs, that quantifies the maximum independent set in tensor powers of a hypergraph. In fact, for multiple problems of hypergraphs of interest, such as the hypergraph corresponding to the capset problem, corner problem, the best known method using the slice rank or the (non-symmetric) subrank cannot give better bounds that one is already known. We hope that the notion of a method based on the symmetric subrank could lead to improved bounds on the capset problem and other combinatorial problems.

This is joint work with Omar Fawzi.