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

The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: SINGn,m{{\rm SING}_{n,m}}, consisting of all m-tuples of n×n{n\times n} complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in SINGn,m{{\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: SINGn,m{{\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 SINGn,m{{\rm SING}_{n,m}}. To prove this result, we identify precisely the group of symmetries of SINGn,m{{\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{m=1}, and suggests a general method for determining the symmetries of algebraic varieties.