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The honeycomb model of $$\text{GL}_n({\mathbb C})$$ tensor products. II: Puzzles determine facets of the Littlewood-Richardson cone. (English) Zbl 1043.05111
In 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 $$V_{N\lambda}\otimes V_{N\mu}$$.

##### 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
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##### References:
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