Concise tensors with maximal symmetries

2026-09-15

AI Summary

解决了确定所有简洁n×n×n张量的最大稳定器维度问题，证明最大值为n²+1，使用了矩阵元组的左右作用上界方法。

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.

Problem

• concise tensors
• stabilizer dimension
• null algebra tensors
• skew symmetric tensor

Innovation

• stabilizer dimension
• concise tensors
• null algebra tensors
• skew symmetric tensor