Permanent versus determinant: not via saturations of monoids of representations

Let Detn denote the closure of the GLn2(C)-orbit of the determinant polynomial detn with respect to linear substitution. The highest weights (partitions) of irreducible GL n 2(C)-representations occurring in the coordinate ring of Detn form a finitely generated monoid S(Detn). We prove that the saturation of S(Detn) contains all partitions λ 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(Detn). AMS subject classifications: 68Q17, 14L24