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 $(\lambda ,\mu ,\nu )$.References
- A. Buch, The saturation conjecture (after A. Knutson and T. Tao), notes from a talk at Berkeley, September 1998.
- Louis J. Billera and Bernd Sturmfels, Fiber polytopes, Ann. of Math. (2) 135 (1992), no. 3, 527–549. MR 1166643, DOI 10.2307/2946575
- A. D. Berenstein and A. V. Zelevinsky, Triple multiplicities for $\textrm {sl}(r+1)$ and the spectrum of the exterior algebra of the adjoint representation, J. Algebraic Combin. 1 (1992), no. 1, 7–22. MR 1162639, DOI 10.1023/A:1022429213282
- A. G. Èlashvili, Invariant algebras, Lie groups, their discrete subgroups, and invariant theory, Adv. Soviet Math., vol. 8, Amer. Math. Soc., Providence, RI, 1992, pp. 57–64. MR 1155664
- W. Fulton, Eigenvalues of sums of Hermitian matrices (after A. Klyachko), Séminaire Bourbaki (1998).
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- O. Gleizer, A. Postnikov, Littlewood-Richardson coefficients via Yang-Baxter equation, in preparation.
- V. Guillemin, C. Zara, Equivariant de Rham theory and graphs, arXiv:math.DG/9808135.
- Alfred Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225–241. MR 140521
- A. Knutson, T. Tao, C. Woodward, The honeycomb model of $\mathrm {GL}_n$ tensor products II: Facets of the L-R cone, in preparation.
- M. M. Kapranov, B. Sturmfels, and A. V. Zelevinsky, Quotients of toric varieties, Math. Ann. 290 (1991), no. 4, 643–655. MR 1119943, DOI 10.1007/BF01459264
- A.A. Klyachko, Stable vector bundles and Hermitian operators, IGM, University of Marne-la-Vallee preprint (1994).
- Shrawan Kumar, Proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture, Invent. Math. 93 (1988), no. 1, 117–130. MR 943925, DOI 10.1007/BF01393689
- Greg Kuperberg, Spiders for rank $2$ Lie algebras, Comm. Math. Phys. 180 (1996), no. 1, 109–151. MR 1403861
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
- A. Zelevinsky, Littlewood-Richardson semigroups, arXiv:math.CO/9704228.
- Günter M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, New York, 1995. MR 1311028, DOI 10.1007/978-1-4613-8431-1
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