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Permanent versus determinant: Not via saturations


Authors: Peter Bürgisser, Christian Ikenmeyer and Jesko Hüttenhain
Journal: Proc. Amer. Math. Soc. 145 (2017), 1247-1258
MSC (2010): Primary 68Q17, 14L24
DOI: https://doi.org/10.1090/proc/13310
Published electronically: November 28, 2016
MathSciNet review: 3589323
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Abstract: Let $ \mathit {Det}_n$ denote the closure of the $ \mathrm {Gl}_{n^2}(\mathbb{C})$-orbit of the determinant polynomial $ \det _n$ with respect to linear substitution. The highest weights (partitions) of irreducible $ \mathrm {Gl}_{n^2}(\mathbb{C})$-representations occurring in the coordinate ring of $ \mathit {Det}_n$ form a finitely generated monoid $ S(\mathit {Det}_n)$. We prove that the saturation of $ S(\mathit {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(\mathit {Det}_n)$.


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Additional Information

Peter Bürgisser
Affiliation: Institute of Mathematics, Technische Universität, Berlin, Germany
Email: pbuerg@math.tu-berlin.de

Christian Ikenmeyer
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 75429
Address at time of publication: Max Planck Institute for Informatics, Saarland Informatics Campus, Germany

Jesko Hüttenhain
Affiliation: Institute of Mathematics, Technische Universität, Berlin, Germany
Email: jesko@math.tu-berlin.de

DOI: https://doi.org/10.1090/proc/13310
Keywords: Geometric complexity theory, permanent versus determinant, representations, saturation of monoids
Received by editor(s): July 9, 2015
Published electronically: November 28, 2016
Additional Notes: The first and third author were partially supported by DFG grant BU 1371/3-2
This research was conducted while the second author was at Texas A&M University.
Communicated by: Harm Derksen
Article copyright: © Copyright 2016 American Mathematical Society