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

We provide counter–examples to Mulmuley’s SH conjecture 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.

[1]  Ernesto Vallejo,et al.  Stability of Kronecker Products of Irreducible Characters of the Symmetric Group , 1999, Electron. J. Comb..

[2]  Ketan Mulmuley,et al.  Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems , 2002, SIAM J. Comput..

[3]  Mercedes H. Rosas.  The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes , 2000 .

[4]  Hariharan Narayanan.  On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients , 2006 .

[5]  Martin E. Dyer,et al.  Sampling contingency tables , 1997 .

[6]  Ketan Mulmuley,et al.  Geometric Complexity III: on deciding positivity of Littlewood-Richardson coefficients , 2005, ArXiv.

[7]  Peter Bürgisser,et al.  The complexity of computing Kronecker coefficients , 2008 .

[8]  D. E. Littlewood,et al.  Products and Plethysms of Characters with Orthogonal, Symplectic and Symmetric Groups , 1958, Canadian Journal of Mathematics.

[9]  Mike Zabrocki,et al.  Expressions for Catalan Kronecker Products , 2008, 0809.3469.

[10]  Hanspeter Kraft,et al.  Classical invariant theory: a primer , 1996 .

[11]  Jeffrey B. Remmel,et al.  On the Kronecker product of Schur functions of two row shapes , 1994 .

[12]  Ketan Mulmuley,et al.  Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties , 2006, SIAM J. Comput..

[13]  M. Brion,et al.  Residue formulae, vector partition functions and lattice points in rational polytopes , 1997 .

[14]  Christophe Tollu,et al.  Stretched Littlewood-Richardson and Kostka Coefficients , 2004 .

[15]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[16]  Etienne Rassart.  A polynomiality property for Littlewood-Richardson coefficients , 2004, J. Comb. Theory, Ser. A.

[17]  F. Murnaghan.  The Analysis of the Kronecker Product of Irreducible Representations of the Symmetric Group , 1938 .

[18]  T. Tao,et al.  The honeycomb model of _{}(ℂ) tensor products I: Proof of the saturation conjecture , 1999 .

[19]  F D Murnaghan.  ON THE ANALYSIS OF THE KRONECKER PRODUCT OF IRREDUCIBLE REPRESENTATIONS OF S(n). , 1955, Proceedings of the National Academy of Sciences of the United States of America.

[20]  T. Tao,et al.  THE SATURATION CONJECTURE (AFTER A. KNUTSON , 2006 .

[21]  Rosa Orellana,et al.  A COMBINATORIAL INTERPRETATION FOR THE COEFFICIENTS IN THE KRONECKER PRODUCT s(n p;p) s , 2005 .

[22]  Emmanuel Briand,et al.  Quasipolynomial formulas for the Kronecker coefficients indexed by two two-row shapes (extended abstract) , 2008 .

[23]  Jesús A. De Loera,et al.  On the Computation of Clebsch–Gordan Coefficients and the Dilation Effect , 2006, Exp. Math..

[24]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[25]  T. Tao,et al.  Honeycombs and sums of Hermitian matrices , 2000, math/0009048.

[26]  Ketan Mulmuley,et al.  Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry , 2007, ArXiv.

[27]  Ketan Mulmuley,et al.  On P vs. NP, Geometric Complexity Theory, and the Flip I: a high level view , 2007, ArXiv.