The honeycomb model of 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

DOI:
https://doi.org/10.1090/S0894-0347-99-00299-4

Published electronically:
April 13, 1999

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

MathSciNet review:
1671451

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Abstract | References | Similar Articles | Additional Information

Abstract: Recently Klyachko has given linear inequalities on triples of dominant weights of necessary for the corresponding Littlewood-Richardson coefficient 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 .

**[Bu]**A. Buch, The saturation conjecture (after A. Knutson and T. Tao), notes from a talk at Berkeley, September 1998.**[BS]**L. Billera, B. Sturmfels, Fiber polytopes,*Annals of Math.*,**135**(1992), no. 3, 527-549. MR**93e:52019****[BZ]**A. Berenstein, A. Zelevinsky, Triple multiplicities for and the spectrum of the exterior algebra of the adjoint representation,*J. Alg. Comb.*,**1**(1992), 7-22. MR**93h:17012****[E]**A.G. Elashvili, Invariant algebras,*Advances in Soviet Math.*,**8**(1992), 57-64. MR**93c:17013****[F]**W. Fulton, Eigenvalues of sums of Hermitian matrices (after A. Klyachko), Séminaire Bourbaki (1998).**[FH]**W. Fulton, J. Harris, Representation theory, Springer-Verlag (1991). MR**93a:20069****[GP]**O. Gleizer, A. Postnikov, Littlewood-Richardson coefficients via Yang-Baxter equation, in preparation.**[GZ]**V. Guillemin, C. Zara, Equivariant de Rham theory and graphs,`math.DG/9808135`.**[H]**A. Horn, Eigenvalues of sums of Hermitian matrices,*Pacific J. Math.*,**12**(1962), 225-241. MR**25:3941****[Hon2]**A. Knutson, T. Tao, C. Woodward, The honeycomb model of tensor products II: Facets of the L-R cone, in preparation.**[KSZ]**M. Kapranov, B. Sturmfels, A. Zelevinsky, Quotients of toric varieties.*Math. Annalen*,**290**(1991), no. 4, 643-655. MR**92g:14050****[Kl]**A.A. Klyachko, Stable vector bundles and Hermitian operators,*IGM, University of Marne-la-Vallee preprint*(1994).**[KMP]**S. Kumar, Proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture,*Invent. math.*,**93**(1988), no. 1, 117-130. MR**89j:17009**O. Mathieu, Construction d'un groupe de Kac-Moody et applications.*Compositio Math.***69**(1989), no. 1, 37-60. MR**90f:17012**P. Polo, Variétés de Schubert et filtrations excellentes.*Astérisque.***10-11**(1989) 281-311. MR**91b:20056****[Ku]**G. Kuperberg, Spiders for rank two Lie algebras,`math.QA/9712143`,*Comm. Math. Phys.*,**180**(1996), no. 1, 109-151. MR**97f:17005****[MFK]**D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory. Third edition. Chapter 8.*Ergebnisse der Mathematik und ihrer Grenzgebiete*, Springer-Verlag, 1994. MR**95m:14012****[Ze]**A. Zelevinsky, Littlewood-Richardson semigroups,`math.CO/9704228`.**[Zi]**G. Ziegler, Lectures on polytopes,*Graduate Texts in Mathematics*, 152. Springer-Verlag, 1995. MR**96a:52011**

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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

Email:
allenk@alumni.caltech.edu

**Terence Tao**

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

Email:
tao@math.ucla.edu

DOI:
https://doi.org/10.1090/S0894-0347-99-00299-4

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