Concise tensors with maximal symmetries

Abstract

Conner, Gesmundo, Landsberg and Ventura (2019) determined the largest stabilizer dimension of concise $n\times n\times n$ tensors that are binding, and they determined the corresponding maximizing tensors to be the null algebra tensors. They left as an open problem to extend this to all concise $n\times n\times n$ tensors (i.e. dropping binding). We solve this problem: We prove that the largest stabilizer dimension of concise $n\times n\times n$ tensors is $n^2+1$ and the maximizers are the null algebra tensors (as in the binding case) and the skew symmetric tensor $e_1\wedge e_2\wedge e_3$. As part of our approach we obtain upper bounds on the stabilizer dimension of matrix tuples under left-right action (generalized Kronecker quiver representations), which we think are of independent interest.