The honeycomb model of \(\text{GL}_n({\mathbb C})\) tensor products. II: Puzzles determine facets of the Littlewood-Richardson cone.

*(English)*Zbl 1043.05111In the first part of this paper [J. Am. Math. Soc. 12, No. 4, 1055–1090 (1999; Zbl 0944.05097)] the authors studied the cone \(\text{BDRY}(n)\) which is the set of triples of weakly decreasing \(n\)-tuples \((\lambda,\mu,\nu)\in ({\mathbb R}^n)^3\) satisfying the three conditions (1) regarding \(\lambda,\mu,\nu\) as spectra of \(n\times n\) Hermitian matrices, there exist three Hermitian matrices with those spectra whose sum is the zero matrix; (2) regarding \(\lambda,\mu,\nu\) as dominant weights of \(\text{GL}_n({\mathbb C})\), the tensor product \(V_{\lambda}\otimes V_{\mu}\otimes V_{\nu}\) of the corresponding irreducible modules contains a \(\text{GL}_n({\mathbb C})\)-invariant vector; (3) regarding \(\lambda,\mu,\nu\) as possible boundary data on a honeycomb, there exist ways to complete it to a honeycomb.

These conditions were proved to be equivalent. A sufficient list of inequalities for this cone was given due to the efforts of several authors: Klyachko, Helmke and Rosenthal, Totaro, and Belkale in terms of Schubert calculus on Grassmannians. In the present, second, part of the paper the authors introduce new combinatorial objects called puzzles, which are certain kinds of diagrams in the triangular lattice in the plane, composed from unit equilateral triangles and unit rhombi, with edges labeled by 0 and 1. Puzzles are used to compute Grassmannian Schubert calculus, and have much interest in their own right. In particular, the authors get new, puzzle-theoretic, proofs of results of Horn and the above-mentioned authors.

The authors also characterize “rigid” puzzles and use them to prove a conjecture of Fulton which states that if the irreducible module \(V_\nu\) appears exactly once in \(V_\lambda \otimes V_\mu\), then for all \(N\in{\mathbb N}\), \(V_{N\lambda}\) appears exactly once in \(V_{N\lambda}\otimes V_{N\mu}\).

These conditions were proved to be equivalent. A sufficient list of inequalities for this cone was given due to the efforts of several authors: Klyachko, Helmke and Rosenthal, Totaro, and Belkale in terms of Schubert calculus on Grassmannians. In the present, second, part of the paper the authors introduce new combinatorial objects called puzzles, which are certain kinds of diagrams in the triangular lattice in the plane, composed from unit equilateral triangles and unit rhombi, with edges labeled by 0 and 1. Puzzles are used to compute Grassmannian Schubert calculus, and have much interest in their own right. In particular, the authors get new, puzzle-theoretic, proofs of results of Horn and the above-mentioned authors.

The authors also characterize “rigid” puzzles and use them to prove a conjecture of Fulton which states that if the irreducible module \(V_\nu\) appears exactly once in \(V_\lambda \otimes V_\mu\), then for all \(N\in{\mathbb N}\), \(V_{N\lambda}\) appears exactly once in \(V_{N\lambda}\otimes V_{N\mu}\).

Reviewer: Vesselin Drensky (Sofia)

##### MSC:

05E05 | Symmetric functions and generalizations |

05E15 | Combinatorial aspects of groups and algebras (MSC2010) |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14N15 | Classical problems, Schubert calculus |

15A42 | Inequalities involving eigenvalues and eigenvectors |

20G05 | Representation theory for linear algebraic groups |

52B12 | Special polytopes (linear programming, centrally symmetric, etc.) |

52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |

05E10 | Combinatorial aspects of representation theory |

##### Keywords:

honeycombs; symmetric functions; Littlewood-Richardson rule; puzzles; Hermitian matrix; eigenvalue problems; Schubert calculus; Grassmannian##### References:

[1] | Prakash Belkale, Local systems on \Bbb P\textonesuperior -\? for \? a finite set, Compositio Math. 129 (2001), no. 1, 67 – 86. · Zbl 1042.14031 · doi:10.1023/A:1013195625868 · doi.org |

[2] | H. Derksen, J. Weyman, On the \(\sigma\)-stable decomposition of quiver representations, preprint available at http://www.math.lsa.umich.edu/ hderksen/preprint.html. · Zbl 1016.16007 |

[3] | H. Derksen, J. Weyman, On the Littlewood-Richardson polynomials, preprint available at http://www.math.lsa.umich.edu/ hderksen/preprint.html. · Zbl 1018.16012 |

[4] | William Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 3, 209 – 249. · Zbl 0994.15021 |

[5] | William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. · Zbl 0878.14034 |

[6] | Oleg Gleizer and Alexander Postnikov, Littlewood-Richardson coefficients via Yang-Baxter equation, Internat. Math. Res. Notices 14 (2000), 741 – 774. · Zbl 0987.20023 · doi:10.1155/S1073792800000416 · doi.org |

[7] | Uwe Helmke and Joachim Rosenthal, Eigenvalue inequalities and Schubert calculus, Math. Nachr. 171 (1995), 207 – 225. · Zbl 0815.15012 · doi:10.1002/mana.19951710113 · doi.org |

[8] | Alfred Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225 – 241. · Zbl 0112.01501 |

[9] | Allen Knutson and Terence Tao, The honeycomb model of \?\?_\?(\?) tensor products. I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (1999), no. 4, 1055 – 1090. · Zbl 0944.05097 |

[10] | A. A. Klyachko, Stable vector bundles and Hermitian operators, IGM, University of Marne-la-Vallee preprint (1994). |

[11] | Allen Knutson, The symplectic and algebraic geometry of Horn’s problem, Linear Algebra Appl. 319 (2000), no. 1-3, 61 – 81. Special Issue: Workshop on Geometric and Combinatorial Methods in the Hermitian Sum Spectral Problem (Coimbra, 1999). · Zbl 0981.15010 · doi:10.1016/S0024-3795(00)00220-2 · doi.org |

[12] | A. Knutson, T. Tao, Puzzles and (equivariant) cohomology of Grassmannians, in preparation. · Zbl 1064.14063 |

[13] | A. Knutson, T. Tao, Puzzles, Littlewood-Richardson rings, and the legend of Procrustes, in preparation. |

[14] | D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. · Zbl 0797.14004 |

[15] | Burt Totaro, Tensor products of semistables are semistable, Geometry and analysis on complex manifolds, World Sci. Publ., River Edge, NJ, 1994, pp. 242 – 250. · Zbl 0873.14016 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.