Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture SH

Details Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture SH Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture SH

2008-10-17

arxiv.org/abs/0810.3163v3

Mathematics

Combinatorics

25 pages. With an appendix by Ketan Mulmuley. To appear in Computational Complexity. See also http://emmanuel.jean.briand.fr

Scientific paper

We provide counter-examples to Mulmuley's strong saturation conjecture (strong SH) for the Kronecker coefficients. This conjecture was proposed in the setting of Geometric Complexity Theory to show that deciding whether or not a Kronecker coefficient is zero can be done in polynomial time. We also provide a short proof of the #P-hardness of computing the Kronecker coefficients. Both results rely on the connections between the Kronecker coefficients and another family of structural constants in the representation theory of the symmetric groups: Murnaghan's reduced Kronecker coefficients. An appendix by Mulmuley introduces a relaxed form of the saturation hypothesis SH, still strong enough for the aims of Geometric Complexity Theory.

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