Permanent versus determinant: not via saturations
arXiv:1501.05528 · doi:10.1090/proc/13310
Abstract
Let Det_n denote the closure of the GL_{n^2}(C)-orbit of the determinant polynomial det_n with respect to linear substitution. The highest weights (partitions) of irreducible GL_{n^2}(C)-representations occurring in the coordinate ring of Det_n form a finitely generated monoid S(Det_n). We prove that the saturation of S(Det_n) contains all partitions lambda with length at most n and size divisible by n. This implies that representation theoretic obstructions for the permanent versus determinant problem must be holes of the monoid S(Det_n).
12 pages; shortened title, corrected error in proof, added bound on stretching factor, provided explicit examples
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