The honeycomb model of GL(n) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone
arXiv:math/0107011
Abstract
The set of possible spectra (λ,μ,ν) 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 λ,μ,νof GL(n,C) the irreducible representation V_νappears exactly once in V_λtensor V_μ, then for all N\in \naturals, V_{Nλ} appears exactly once in V_{Nλ} tensor V_{Nμ}.''
arXiv commentFinal version. Most significant change is the inclusion of a proof that puzzles do indeed compute Schubert calculus (before, this had been intended to be left to a separate paper)
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- Proof of Stembridge's conjecture on stability of Kronecker coefficients
- Geometric Proofs of Horn and Saturation Conjectures
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- The Horn inequalities from a geometric point of view
- Generalised Stretched Littlewood-Richardson Coefficients
- Concave cocirculations in a triangular grid