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The honeycomb model of $GL_n({\mathbb C})$ tensor products II: Puzzles determine facets of the Littlewood-Richardson cone

Authors: Allen Knutson, Terence Tao and Christopher Woodward
Journal: J. Amer. Math. Soc. 17 (2004), 19-48
MSC (2000): Primary 14N15; Secondary 15A42, 52B12, 05E10
Published electronically: October 14, 2003
Part I: J. Amer. Math. Soc. 12 (1999), 1055-1090
MathSciNet review: 2015329
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Abstract: The set of possible spectra $(\lambda,\mu,\nu)$ of zero-sum triples of Hermitian matrices forms a polyhedral cone, whose facets have been already studied by Knutson and Tao, Helmke and Rosenthal, Totaro, and Belkale in terms of Schubert calculus on Grassmannians. We give a complete determination of these facets; there is one for each triple of Grassmannian Schubert cycles intersecting in a unique point. In particular, the list of inequalities determined by Belkale to be sufficient is in fact minimal.

We introduce puzzles, which are new combinatorial gadgets to compute Grassmannian Schubert calculus, and seem to have much interest in their own right. As the proofs herein indicate, the Hermitian sum problem is very naturally studied using puzzles directly, and their connection to Schubert calculus is quite incidental to our approach. In particular, we get new, puzzle-theoretic, proofs of the results of Horn, Klyachko, Helmke and Rosenthal, Totaro, and Belkale.

Along the way we give a characterization of ``rigid'' puzzles, which we use to prove a conjecture of W. Fulton: ``if for a triple of dominant weights $\lambda,\mu,\nu$ of $GL_n({\mathbb C})$ the irreducible representation $V_\nu$appears exactly once in $V_\lambda \otimes V_\mu$, then for all $N\in {\mathbb N}$, $V_{N\lambda}$ appears exactly once in $V_{N\lambda}\otimes V_{N\mu}$.''

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Additional Information

Allen Knutson
Affiliation: Mathematics Department, UC Berkeley, Berkeley, California 94720

Terence Tao
Affiliation: Mathematics Department, UCLA, Los Angeles, California 90095-1555

Christopher Woodward
Affiliation: Mathematics Department, Rutgers University, New Brunswick, New Jersey 08854-8019

Received by editor(s): July 2, 2001
Published electronically: October 14, 2003
Additional Notes: The first author was supported by an NSF Postdoctoral Fellowship, an NSF grant, and the Clay Mathematics Institute
The second author was supported by the Clay Mathematics Institute, and grants from the Sloan and Packard Foundations
The third author was partially supported by an NSF Postdoctoral Fellowship, and NSF Grant 9971357
Article copyright: © Copyright 2003 American Mathematical Society