Geodesic
The honeycomb model of 𝐺𝐿_{𝑛}(ℂ) tensor products I: Proof of the saturation conjecture
Knutson, Allen 1 ,  2   ; Tao, Terence 3
1 Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254
2 Department of Mathematics, University of California Berkeley, Berkeley, California 94720-3840
3 Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095-1555
Journal of the American Mathematical Society, Tome 12 (1999) no. 4, pp. 1055-1090 Cet article a éte moissonné depuis la source American Mathematical Society

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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 $(\lambda ,\mu ,\nu )$.
DOI : 10.1090/S0894-0347-99-00299-4

Knutson, Allen  1 , 2   ; Tao, Terence  3

1 Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254
2 Department of Mathematics, University of California Berkeley, Berkeley, California 94720-3840
3 Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095-1555 
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Knutson, Allen; Tao, Terence. The honeycomb model of 𝐺𝐿_{𝑛}(ℂ) tensor products I: Proof of the saturation conjecture. Journal of the American Mathematical Society, Tome 12 (1999) no. 4, pp. 1055-1090. doi: 10.1090/S0894-0347-99-00299-4

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