Singular tuples of matrices is not a null cone (and, the symmetries of algebraic varieties)
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The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: SING_n,m, consisting of all m-tuples of n× n complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in 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: 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 SING_n,m. To prove this result we identify precisely the group of symmetries of 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.
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