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The honeycomb model of $GL_n(\mathbb{C})$ tensor products I:
Proof of the saturation conjecture

Authors: Allen Knutson and Terence Tao
Journal: J. Amer. Math. Soc. 12 (1999), 1055-1090
MSC (1991): Primary 05E15, 22E46; Secondary 15A42
Published electronically: April 13, 1999
Part II: J. Amer. Math. Soc. (2004), 19-48
MathSciNet review: 1671451
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Abstract: Recently Klyachko has given linear inequalities on triples $(\lambda,\mu,\nu)$ of dominant weights of $GL_n(\mathbb{C})$ necessary for the corresponding Littlewood-Richardson coefficient $\dim (V_\lambda \otimes V_\mu \otimes V_\nu)^{GL_n(\mathbb{C})}$ to be positive. We show that these conditions are also sufficient, which was known as the saturation conjecture. In particular this proves Horn's conjecture from 1962, giving a recursive system of inequalities. Our principal tool is a new model of the Berenstein-Zelevinsky cone for computing Littlewood-Richardson coefficients, the honeycomb model. The saturation conjecture is a corollary of our main result, which is the existence of a particularly well-behaved honeycomb associated to regular triples $(\lambda,\mu,\nu)$.

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Additional Information

Allen Knutson
Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254
Address at time of publication: Department of Mathematics, University of California Berkeley, Berkeley, California 94720-3840

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

Keywords: Honeycombs, Littlewood-Richardson coefficients, Berenstein-Zelevinsky patterns, Horn's conjecture, saturation, Klyachko inequalities
Received by editor(s): July 31, 1998
Received by editor(s) in revised form: February 25, 1999
Published electronically: April 13, 1999
Additional Notes: The first author was supported by an NSF Postdoctoral Fellowship.
The second author was partially supported by NSF grant DMS-9706764.
Article copyright: © Copyright 1999 American Mathematical Society