Summary: 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 selfadjoint elements \(a,b,c\) with \(a+b+c=0\) in the factor \(\mathcal R^{\omega}\) 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 selfadjoint elements of such a factor and \(x+y+z=0\), then there exist selfadjoint \(a,b,c \in \mathcal R^{\omega}\) 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 this result is known to imply an affirmative answer to an embedding question formulated by Connes.
The critical point in the proof of this result is the production of elements in the intersection of three Schubert varieties. When the factor under consideration is the algebra of \(n\times n\) complex matrices, our arguments provide new and elementary proofs of the Horn inequalities, which do not require knowledge of the structure of the cohomology of the Grassmann manifolds.