Deciding Positivity of Littlewood-Richardson Coefficients

Starting with Knutson and Tao's hive model [J. Amer. Math. Soc., 12 (1999), pp. 1055--1090] we characterize the Littlewood--Richardson coefficient ${c_{\lambda,\mu}^{\nu}}$ of given partitions $\lambda,\mu,\nu\in\mathbb{N}^n$ as the number of capacity achieving hive flows on the honeycomb graph. Based on this, we design a polynomial time algorithm for deciding ${c_{\lambda,\mu}^{\nu}} >0$. This algorithm is easy to state and takes $\mathcal{O}(n^3\log\nu_1)$ arithmetic operations and comparisons. We further show that the capacity achieving hive flows can be seen as the vertices of a connected graph, which leads to new structural insights into Littlewood--Richardson coefficients.

[1]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

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

[3]  Ketan Mulmuley,et al.  On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna , 2011, JACM.

[4]  A. Karimi,et al.  Master‟s thesis , 2011 .

[5]  U. Helmke,et al.  Eigenvalue inequalities and Schubert calculus , 1995 .

[6]  Andrei Zelevinsky,et al.  Triple Multiplicities for sl(r + 1) and the Spectrum of the Exterior Algebra of the Adjoint Representation , 1992 .

[7]  B. M. Fulk.  MATH , 1992 .

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

[9]  Greta Panova,et al.  On the complexity of computing Kronecker coefficients , 2014, computational complexity.

[10]  Peter Bürgisser,et al.  A max-flow algorithm for positivity of Littlewood-Richardson coefficients , 2009 .

[11]  Eric V. Denardo,et al.  Flows in Networks , 2011 .

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

[13]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[14]  W. Fulton.  Eigenvalues, invariant factors, highest weights, and Schubert calculus , 1999, math/9908012.

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

[16]  Felipe Cucker,et al.  Condition - The Geometry of Numerical Algorithms , 2013, Grundlehren der mathematischen Wissenschaften.

[17]  F. Eisenbrand,et al.  On the complexity of computing Kronecker coecients and deciding positivity of Littlewood-Richardson coecients , 2008 .

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

[19]  Igor Pak,et al.  Combinatorics and geometry of Littlewood-Richardson cones , 2005, Eur. J. Comb..

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

[21]  E. Gentile.  On the invariant factors , 1976 .

[22]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[23]  Terence Tao,et al.  The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture , 1998, math/9807160.

[24]  A. Klyachko.  Stable bundles, representation theory and Hermitian operators , 1998 .

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

[26]  Christian Ikenmeyer.  2 Flow description of LR coefficients 2 . 1 Flows on digraphs , 2012 .

[27]  D MulmuleyKetan.  On P vs. NP and geometric complexity theory , 2011 .

[28]  T. Tao,et al.  The honeycomb model of _{}(ℂ) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone , 2001, math/0107011.

[29]  Anders S. Buch.  The saturation conjecture (after A. Knutson and T. Tao) , 1998 .