## 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 - DOI: https://doi.org/10.1090/S0894-0347-99-00299-4
- Published electronically: April 13, 1999
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Part II: J. Amer. Math. Soc. (2004), 19-48

## 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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## 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: allenk@alumni.caltech.edu
**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: tao@math.ucla.edu
- 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: https://doi.org/10.1090/S0894-0347-99-00299-4
- MathSciNet review: 1671451