Geodesic
The honeycomb model of 𝐺𝐿_{𝑛}(ℂ) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone
Knutson, Allen1 ; Tao, Terence2 ; Woodward, Christopher3
1 Mathematics Department, UC Berkeley, Berkeley, California 94720
2 Mathematics Department, UCLA, Los Angeles, California 90095-1555
3 Mathematics Department, Rutgers University, New Brunswick, New Jersey 08854-8019
Journal of the American Mathematical Society, Tome 17 (2004) no. 1, pp. 19-48

Voir la notice de l'article provenant de la source American Mathematical Society

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 “rigid” puzzles, which we use to prove a conjecture of W. Fulton: “if 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 $V_{N\lambda }\otimes V_{N\mu }$.”
DOI : 10.1090/S0894-0347-03-00441-7

Knutson, Allen 1 ; Tao, Terence 2 ; Woodward, Christopher 3

1 Mathematics Department, UC Berkeley, Berkeley, California 94720
2 Mathematics Department, UCLA, Los Angeles, California 90095-1555
3 Mathematics Department, Rutgers University, New Brunswick, New Jersey 08854-8019 
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Knutson, Allen; Tao, Terence; Woodward, Christopher. The honeycomb model of 𝐺𝐿_{𝑛}(ℂ) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone. Journal of the American Mathematical Society, Tome 17 (2004) no. 1, pp. 19-48. doi: 10.1090/S0894-0347-03-00441-7

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