Buergisser, Peter, Ikenmeyer, Christian and Panova, Greta
(2019)
No occurrence obstructions in geometric complexity theory.
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, 32 (1).
pp. 163-193.
Abstract
<p>The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper V normal upper P Subscript normal w normal s"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">V</mml:mi> <mml:mi mathvariant="normal">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">w</mml:mi> <mml:mi mathvariant="normal">s</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathrm {VP}_{\mathrm {ws}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper V normal upper N normal upper P"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">V</mml:mi> <mml:mi mathvariant="normal">N</mml:mi> <mml:mi mathvariant="normal">P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {VNP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In 2001 Mulmuley and Sohoni suggested studying a strengthened version of this conjecture over complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting <italic>occurrence obstructions</italic>, which are irreducible representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G normal upper L Subscript n squared Baseline left-parenthesis double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">G</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {GL}_{n^2}(\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via <italic>multiplicity obstructions</italic> as proposed by Mulmuley and Sohoni in 2001.</p>
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | Permanent versus determinant, geometric complexity theory, orbit closures, representations, plethysms, Young tableaux, tensors, highest weight vectors |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 20 Jan 2020 09:45 |
| Last Modified: | 19 Jan 2023 00:12 |
| DOI: | 10.1090/jams/908 |
| Open Access URL: | https://arxiv.org/pdf/1604.06431 |
| Related URLs: | |
| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3067060 |
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