Singular tuples of matrices is not a null cone (and, the symmetries of algebraic varieties)

TL;DR

This paper proves that the algebraic variety of m-tuples of singular matrices is not a null cone of any reductive group action, revealing structural complexity and identifying its symmetry group.

Contribution

It characterizes the symmetry group of the variety of singular matrix tuples and shows this variety is not a null cone, contrasting with non-commutative cases.

Findings

${ m SING}_{n,m}$ is not a null cone of any reductive group action

The symmetry group of ${ m SING}_{n,m}$ is precisely identified

The work generalizes Frobenius's classical result for $m=1$

Abstract

The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: ${\rm SING}_{n,m}$, consisting of all $m$-tuples of $n\times n$ complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in ${\rm SING}_{n,m}$ will imply super-polynomial circuit lower bounds, a holy grail of the theory of computation. A sequence of recent works suggests such efficient algorithms for memberships in a general class of algebraic varieties, namely the null cones of linear group actions. Can this be used for the problem above? Our main result is negative: ${\rm SING}_{n,m}$ is not the null cone of any (reductive) group action! This stands in stark contrast to a non-commutative analog of this variety, and points to an inherent structural difficulty of ${\rm SING}_{n,m}$. To prove this result we identify precisely the group of symmetries of ${\rm SING}_{n,m}$. We find this characterization, and the tools we introduce to prove it, of independent interest. Our work significantly generalizes a result of Frobenius for the special case $m=1$, and suggests a general method for determining the symmetries of algebraic varieties.