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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
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 $(\lambda,\mu,\nu)$ 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 $(\lambda,\mu,\nu)$.


References [Enhancements On Off] (What's this?)

  • [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 $sl(r+1)$ 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 ${{GL_n(\mathbb{C})}}$ 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

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