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
Full-text PDF Free Access
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 $(\lambda ,\mu ,\nu )$.
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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
MR Author ID:
361755
ORCID:
0000-0002-0140-7641
Email:
tao@math.ucla.edu
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