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

Abstract

The set of possible spectra ( λ , μ , ν ) (\lambda ,\mu ,\nu ) of zero-sum triples of Hermitian matrices forms a polyhedral cone, whose facets have been already studied by Knutson and Tao, Helmke and Rosenthal, Totaro, and Belkale in terms of Schubert calculus on Grassmannians. We give a complete determination of these facets; there is one for each triple of Grassmannian Schubert cycles intersecting in a unique point. In particular, the list of inequalities determined by Belkale to be sufficient is in fact minimal. We introduce puzzles, which are new combinatorial gadgets to compute Grassmannian Schubert calculus, and seem to have much interest in their own right. As the proofs herein 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 of Horn, Klyachko, Helmke and Rosenthal, Totaro, and Belkale. 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 , ν \lambda ,\mu ,\nu of G L n ( C ) GL_n({\mathbb C}) the irreducible representation V ν V_\nu appears exactly once in V λ ⊗ V μ V_\lambda \otimes V_\mu , then for all N ∈ N N\in {\mathbb N} , V N λ V_{N\lambda } appears exactly once in V N λ ⊗ V N μ V_{N\lambda }\otimes V_{N\mu } .”

A Geometric Description of Feasible Singular Values in the Tensor Train Format

“…The following result by Knutson and Tao poses a complete resolution to Weyl's problem and is based on preceding breakthroughs [16,20,22,24]. This problem has since then also been generalized, for example [9,11].…”

“…Considerations for simple chains of honeycombs (cf. Lemma 4.6) have also been made in [22,24], but we need to rephrase these ideas for our own purposes. We define the hive set HIVE n,M (∼ S ) as set of all (n, M )-hives H with structure ∼ S .…”

A normal variation of the Horn problem: the rank 1 case

“…In [2] the uniticity in Theorem 3.1 was proven by using a result by Hochstadt [4], which gives uniqueness of a tridiagonal Hermitian matrix with prescribed different eigenvalues for the full matrix and its principal submatrix of one size smaller. The uniqueness for the full Horn problem is addressed in a corollary in the appendix of [8], where the result is stated in terms of honeycombs.…”

Многогранник Агнихотри - Вудварда - Белкале И Конусы Клячко

“…Доказательство. Следует из предложений 1 и 2, а также из результата Кнутсона, Тао и Вудварда [13] о минимальности системы неравенств {h t 0} t∈I0 , задающих K .…”