Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians

Abstract

ABSTRACT. The saturation theorem of We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland's problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aformentioned work together with .

Vanishing of Littlewood–Richardson polynomials is in P

“…Moreover, by our construction, NP ν λ,µ = P N ν N λ,N µ , which means C N ν N λ,N µ = 0. Thus, by a saturation theorem of D. Anderson, E. Richmond, and the third author [1], we conclude C ν λ,µ = 0 ⇐⇒ C N ν N λ,N µ = 0 ⇐⇒ P ν λ,µ = ∅. To determine if P ν λ,µ = ∅, one needs to ascertain feasiblity of any linear programming problem involving P ν λ,µ .…”

“…By (2), and/or the sentence immediately after it, there must be an "extra" edge label k in row i + 1 2 and in column j or to its left. Case 1: [T i,j < k] The rightmostness of the placement of the k's (see (1)) implies that k ∈ T i+ 1 2 ,j . Hence, the row word has more k + 1's than k's before reading edge (i + 1 2 , j), a contradiction of the rightmostness of k + 1.…”

“…One cannot use the same general method of this paper to resolve Question 1. To be precise, in [5,Section 7] it is noted that (up to a sign convention) k (2,1) (1,0),(1,0) = 1 but k (4,2) (2,0),(2,0) = 0. That is, the saturation statement k ν λ,µ = 0 =⇒ k N ν N λ,N µ = 0 is false (the truth of the converse is not known).…”

Extremal rays of the equivariant Littlewood-Richardson cone

“…In this work we collide the two worlds of [Bel19] and [ARY13] in order to answer a question of C. Robichaux, H. Yadav, and A. Yong [RYY]. In [Bel19], P. Belkale introduced an algorithm for finding the extremal rays of the Hermitian eigencone (also called the tensor cone or Littlewood-Richardson cone) -the pointed rational cone which among other things governs the nonvanishing of Littlewood-Richardson coefficients.…”

“…In [Bel19], P. Belkale introduced an algorithm for finding the extremal rays of the Hermitian eigencone (also called the tensor cone or Littlewood-Richardson cone) -the pointed rational cone which among other things governs the nonvanishing of Littlewood-Richardson coefficients. In [ARY13], D. Anderson, E. Richmond, and A. Yong proved that the equivariant Littlewood-Richardson nonvanishing problem is determined by a similar cone, of which the former is a facet, thereby proving the equivariant nonvanishing problem to be saturated. Here, we naturally adapt Belkale's algorithm to the equivariant setting, repeatedly making use of the core Proposition 2.1 from [ARY13], thus finding most of the extremal rays of the equivariant Littlewood-Richardson cone.…”