The honeycomb model of _{}(ℂ) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone

By Allen Knutson, Terence Tao, Christopher Woodward. 2001

284 citations recorded by the source corpus.

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}.''

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a2949589bcd6423302040793c6b7fb18bf95db8a

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Citations in the corpus

Geometric invariant theory and the generalized eigenvalue problem (2007, methodology): “tained the same inequalities as in Theorem A in [BK06] but without proving that this list is optimal. In fact Theorem A were known in some particular cases. Indeed, Knutson, Tao and Woodward shown in [KTW04] the case when G = SLn is diagonally embedded in SLn × SLn by combinatorial tools. Using the ”
The many aspects of counting lattice points in polytopes (2005, unknown): “[16,21,41, 42 ]). One example of such polytopes comes from the Gelfand-Tsetlin patterns which arise in the representation theory of gl n C .W hat is a Gelfand Tsetlin pattern? It is a triangular array of numbers x = (xij)1≤i≤j ≤n satisfying that xij ≥ 0, for 1 ≤ i ≤ j ≤ n ;a ndxi,j +1 ≥ xij ≥ xi+1,j”
A Littlewood–Richardson rule for two-step flag varieties (2009, background): “For instance, in recent years, new Littlewood-Richardson rules for the Grassmannians have enabled Klyachko, Knutson, Tao, Woodward and their collaborators to resolve long standing problems such as Horn’s Conjecture and the Saturation Conjecture (see [KT] and [ KTW ]).”
A k-tableau characterization of k-Schur functions (2005, background): “Thus, there should exist a pair of statistics on k-standard tableaux to explain the k-Schur expansion coefficients in a Macdonald polynomial Hµ[X ; q, t] since h1n = Hµ[X ; 1, 1].”
QUANTUM COHOMOLOGY AND THE k-SCHUR BASIS (2005, background): “Although there has been progress in certain cases [2, 3, 4, 13, 14 , 25, 26, 28], a combinatorial interpretation for the 3-point Gromov-Witten invariants in complete generality remains an open problem.”
Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity (2003, background): “In the case n = 3 all inequalities are weak triangle inequalities and moreover the system of inequalities is known to be irredundant, see [ KTW ].”
Eigenvalue problem and a new product in cohomology of flag varieties (2004, background): “Knutson-Tao-Woodward [ KTW ] proved that for G = SL(n) the inequalities (B) (which are, in this case, the same set of inequalities as (B� ) because of Lemma 19) are irredundant.”