Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor

Abstract

The intersection ring of a complex Grassmann manifold is generated by Schubert varieties, and its structure is governed by the Littlewood-Richardson rule. Given three Schubert varieties S 1 , S 2 , S 3 with intersection number equal to one, we show how to construct an explicit element in their intersection. This element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags, and is therefore well defined over an arbitrary field of scalars. Moreover, this result also applies to appropriately defined analogues of Schubert varieties in the Grassmann manifolds associated with a finite von Neumann algebra. The arguments require the combinatorial structure of honeycombs, particularly the structure of the rigid extremal honeycombs. It is known that the eigenvalue distributions of self-adjoint elements a, b, c with a + b + c = 0 in the factor R ω are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. In particular, if x, y, z are self-adjoint elements of such a factor and x + y + z = 0, then there exist self-adjoint a, b, c ∈ R ω such that a + b + c = 0 and a (respectively, b, c) has the same eigenvalue distribution as x (respectively, y, z). A ('complete') matricial form of

A family of reductions for Schubert intersection problems

“…Our proofs deepen some of the results in [1]. Even though we review the relevant results of [1], familiarity with that paper would be helpful in reading this one.…”

“…3 we discuss a special class of measures, the tree measures. It was implicit in the results of [1] that rigid extremal measures have an underlying tree structure, and this is made explicit here. Section 4 reviews the construction of a puzzle from a measure, and uses the results of Sect.…”

“…Even though we review the relevant results of [1], familiarity with that paper would be helpful in reading this one. The paper benefited from a careful reading by the referees who helped improve the exposition, and suggested that this work is connected to earlier results about the factorization of Littlewood-Richardson coefficients [5,7].…”

“…2 we describe the formulation of the Littlewood-Richardson rule in terms of measures. This is essentially the puzzle formulation of [8], and was also used in [1]. We also introduce the linear combinations of dimensions which serve as witnesses for the possibility of reductions.…”

“…Thus, the set M r is a convex polyhedral cone, and therefore each measure 0 = m ∈ M r can be written as a sum of extremal measures. Recall that m = 0 is extremal if every measure m ≤ m is a multiple of m. This decomposition into extremal summands is unique (except for the order of the terms) if m is a rigid measure (see [1,Corollary 3.6]). In the proof of Theorem 3.1, we will describe briefly the result of [1] showing how the extremal summands of a rigid measure are obtained.…”