The honeycomb model of $GL_n({\mathbb C})$ tensor products II: Puzzles determine facets of the Littlewood-Richardson cone

Authors:
Allen Knutson, Terence Tao and Christopher Woodward

Journal:
J. Amer. Math. Soc. **17** (2004), 19-48

MSC (2000):
Primary 14N15; Secondary 15A42, 52B12, 05E10

DOI:
https://doi.org/10.1090/S0894-0347-03-00441-7

Published electronically:
October 14, 2003

Part I:
J. Amer. Math. Soc. **12** (1999), 1055-1090

MathSciNet review:
2015329

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The set of possible spectra $(\lambda ,\mu ,\nu )$ of zero-sum triples of Hermitian matrices forms a polyhedral cone, whose facets have been already studied by Knutson and Tao, Helmke and Rosenthal, Totaro, and Belkale in terms of Schubert calculus on Grassmannians. We give a complete determination of these facets; there is one for each triple of Grassmannian Schubert cycles intersecting in a unique point. In particular, the list of inequalities determined by Belkale to be sufficient is in fact minimal. We introduce *puzzles*, which are new combinatorial gadgets to compute Grassmannian Schubert calculus, and seem to have much interest in their own right. As the proofs herein indicate, the Hermitian sum problem is very naturally studied using puzzles directly, and their connection to Schubert calculus is quite incidental to our approach. In particular, we get new, puzzle-theoretic, proofs of the results of Horn, Klyachko, Helmke and Rosenthal, Totaro, and Belkale. Along the way we give a characterization of βrigidβ puzzles, which we use to prove a conjecture of W. Fulton: βif for a triple of dominant weights $\lambda ,\mu ,\nu$ of $GL_n({\mathbb C})$ the irreducible representation $V_\nu$ appears exactly once in $V_\lambda \otimes V_\mu$, then for all $N\in {\mathbb N}$, $V_{N\lambda }$ appears exactly once in $V_{N\lambda }\otimes V_{N\mu }$.β

- Prakash Belkale,
*Local systems on $\Bbb P^1-S$ for $S$ a finite set*, Compositio Math.**129**(2001), no. 1, 67β86. MR**1856023**, DOI https://doi.org/10.1023/A%3A1013195625868
[DW1]DW1 H. Derksen, J. Weyman, On the $\sigma$-stable decomposition of quiver representations, preprint available at http://www.math.lsa.umich.edu/~hderksen/preprint.html.
[DW2]DW2 H. Derksen, J. Weyman, On the Littlewood-Richardson polynomials, preprint available at http://www.math.lsa.umich.edu/~hderksen/preprint.html.
- William Fulton,
*Eigenvalues, invariant factors, highest weights, and Schubert calculus*, Bull. Amer. Math. Soc. (N.S.)**37**(2000), no. 3, 209β249. MR**1754641**, DOI https://doi.org/10.1090/S0273-0979-00-00865-X - William Fulton,
*Young tableaux*, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR**1464693** - Oleg Gleizer and Alexander Postnikov,
*Littlewood-Richardson coefficients via Yang-Baxter equation*, Internat. Math. Res. Notices**14**(2000), 741β774. MR**1776619**, DOI https://doi.org/10.1155/S1073792800000416 - Uwe Helmke and Joachim Rosenthal,
*Eigenvalue inequalities and Schubert calculus*, Math. Nachr.**171**(1995), 207β225. MR**1316359**, DOI https://doi.org/10.1002/mana.19951710113 - Alfred Horn,
*Eigenvalues of sums of Hermitian matrices*, Pacific J. Math.**12**(1962), 225β241. MR**140521** - Allen Knutson and Terence Tao,
*The honeycomb model of ${\rm GL}_n({\bf C})$ tensor products. I. Proof of the saturation conjecture*, J. Amer. Math. Soc.**12**(1999), no. 4, 1055β1090. MR**1671451**, DOI https://doi.org/10.1090/S0894-0347-99-00299-4
[Kl]Kly A. A. Klyachko, Stable vector bundles and Hermitian operators, IGM, University of Marne-la-Vallee preprint (1994).
- Allen Knutson,
*The symplectic and algebraic geometry of Hornβs problem*, Linear Algebra Appl.**319**(2000), no. 1-3, 61β81. Special Issue: Workshop on Geometric and Combinatorial Methods in the Hermitian Sum Spectral Problem (Coimbra, 1999). MR**1799624**, DOI https://doi.org/10.1016/S0024-3795%2800%2900220-2
[KT1]Puz1 A. Knutson, T. Tao, Puzzles and (equivariant) cohomology of Grassmannians, in preparation.
[KT2]Puz2 A. Knutson, T. Tao, Puzzles, Littlewood-Richardson rings, and the legend of Procrustes, in preparation.
- 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** - Burt Totaro,
*Tensor products of semistables are semistable*, Geometry and analysis on complex manifolds, World Sci. Publ., River Edge, NJ, 1994, pp. 242β250. MR**1463972**

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Additional Information

**Allen Knutson**

Affiliation:
Mathematics Department, UC Berkeley, Berkeley, California 94720

Email:
allenk@math.berkeley.edu

**Terence Tao**

Affiliation:
Mathematics Department, UCLA, Los Angeles, California 90095-1555

MR Author ID:
361755

ORCID:
0000-0002-0140-7641

Email:
tao@math.ucla.edu

**Christopher Woodward**

Affiliation:
Mathematics Department, Rutgers University, New Brunswick, New Jersey 08854-8019

MR Author ID:
603893

Email:
ctw@math.rutgers.edu

Received by editor(s):
July 2, 2001

Published electronically:
October 14, 2003

Additional Notes:
The first author was supported by an NSF Postdoctoral Fellowship, an NSF grant, and the Clay Mathematics Institute

The second author was supported by the Clay Mathematics Institute, and grants from the Sloan and Packard Foundations

The third author was partially supported by an NSF Postdoctoral Fellowship, and NSF Grant 9971357

Article copyright:
© Copyright 2003
American Mathematical Society