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Permanent Versus Determinant: Not Via Saturations
Citation key BIH-Permanent-Versus-Determinant-Not-Via-Saturations
Author Peter Bürgisser and Christian Ikenmeyer and Jesko Hüttenhain
Pages 1247-1258
Year 2016
DOI 10.1090/proc/13310
Journal Proc. AMS
Volume 145
Month 11
Abstract Let $Det_n$ denote the closure of the $\mathrm G\mathrm L(n^2,\mathbb C)$-orbit of the determinant polynomial $\det_n$ with respect to linear substitution. The highest weights (partitions) of irreducible $\mathrm G\mathrm L(n^2,\mathbb C)$-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 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 $S(Det_n)$.
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