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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Permanent versus determinant: Not via saturations
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by Peter Bürgisser, Christian Ikenmeyer and Jesko Hüttenhain PDF
Proc. Amer. Math. Soc. 145 (2017), 1247-1258 Request permission

Abstract:

Let $Det_n$ denote the closure of the $\mathrm {Gl}_{n^2}(\mathbb {C})$-orbit of the determinant polynomial $\mathrm {det}_n$ with respect to linear substitution. The highest weights (partitions) of irreducible $\mathrm {Gl}_{n^2}(\mathbb {C})$-representations occurring in the coordinate ring of $Det_n$ 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 $S(Det_n)$.
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Additional Information
  • Peter Bürgisser
  • Affiliation: Institute of Mathematics, Technische Universität, Berlin, Germany
  • MR Author ID: 316251
  • 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
  • MR Author ID: 911976
  • Jesko Hüttenhain
  • Affiliation: Institute of Mathematics, Technische Universität, Berlin, Germany
  • Email: jesko@math.tu-berlin.de
  • 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
  • © Copyright 2016 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 1247-1258
  • MSC (2010): Primary 68Q17, 14L24
  • DOI: https://doi.org/10.1090/proc/13310
  • MathSciNet review: 3589323