Mathematics – Combinatorics
Scientific paper
2001-07-02
Journal of the AMS, 17 (2004) 19-48
Mathematics
Combinatorics
Final version. Most significant change is the inclusion of a proof that puzzles do indeed compute Schubert calculus (before, t
Scientific paper
The set of possible spectra (\lambda,\mu,\nu) of zero-sum triples of Hermitian matrices forms a polyhedral cone. We give a complete determination of its facets, finishing a long story with recent highlights by [Helmke-Rosenthal, Klyachko, Belkale]. We introduce_puzzles_, which are new combinatorial gadgets to compute Grassmannian Schubert calculus, and will probably be the main point of interest for many readers. As the proofs 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 in [H,Kly,HR,Be]. 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,C) the irreducible representation V_\nu appears exactly once in V_\lambda tensor V_\mu, then for all N\in \naturals, V_{N\lambda} appears exactly once in V_{N\lambda} tensor V_{N\mu}.''
Knutson Allen
Tao Terence
Woodward Christopher
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