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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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The honeycomb model of $GL_n(\mathbb C)$ tensor products I: Proof of the saturation conjecture
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by Allen Knutson and Terence Tao
J. Amer. Math. Soc. 12 (1999), 1055-1090
Published electronically: April 13, 1999

Part II: J. Amer. Math. Soc. (2004), 19-48


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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Bibliographic 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
  • Email:
  • Terence Tao
  • Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095-1555
  • MR Author ID: 361755
  • ORCID: 0000-0002-0140-7641
  • Email:
  • 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.
  • © Copyright 1999 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 12 (1999), 1055-1090
  • MSC (1991): Primary 05E15, 22E46; Secondary 15A42
  • DOI:
  • MathSciNet review: 1671451