On a reduction procedure for Horn inequalities in finite von Neumann algebras

Abstract

Abstract. We consider the analogues of the Horn inequalities in finite von Neumann algebras, which concern the possible spectral distributions of sums a + b of self-adjoint elements a and b in a finite von Neumann algebra. It is an open question whether all of these Horn inequalities must hold in all finite von Neumann algebras, and this is related to Connes' embedding problem. For each choice of integers 1 ≤ r ≤ n, there is a set T n r of Horn triples (I, J, K) of r-tuples of integers, and the Horn inequalities are in one-to-one correspondence with ∪ 1≤r≤n T n r . We consider a property P n , analogous to one introduced by Therianos and Thompson in the case of matrices, amounting to the existence of projections having certain properties relative to arbitrary flags, which guarantees that a given Horn inequality holds in all finite von Neumann algebras. It is an open question whether all Horn triples in T n r have property P n . Certain triples in T n r can be reduced to triples in T n−1 r by an operation we call TT-reduction. We show that property P n holds for the original triple if property P n−1 holds for the reduced one. A major part of this paper is devoted to showing that this operation of reduction preserves the value of the corresponding Littlewood-Richardson coefficients. We then characterize the TT-irreducible Horn triples in T n 3 , for arbitrary n, and for those LR-minimal ones (namely, those having Littlewood-Richardson coefficient equal to 1), we perform a construction of projections with respect to flags in arbitrary von Neumann algebras in order to prove property P n for them. This shows that all LR-minimal triples in ∪ n≥3 T n 3 have property P n , and so that the corresponding Horn inequalities hold in all finite von Neumann algebras.

A family of reductions for Schubert intersection problems

“…As in the result of [3] just mentioned, we have c I J K = c I J K . The general reduction we propose can be described as follows.…”

“…The question arises naturally whether c I J K = 0 if c I J K = 0, so that the reduced problem is still guaranteed to have a solution. That this is indeed the case was shown by Collins and Dykema [3] who proved that in fact c I J K = c I J K .…”

“…This is equivalent with the puzzle description of [8]. Choose unit vectors w 1 , w 2 , w 3 in the plane such that w 1 + w 2 + w 3 = 0.…”

“…The following figure illustrates the process as applied to a measure m = m ϕ whose support is pictured below, and ν is the measure whose inflation was depicted in the preceding figure. We have oriented all the edges away from the branch point inside 3 , and completed the outline of 6 .…”

“…Let e 1 , e 2 , e 3 be three edges of T adjacent to a vertex V , and assume that e 1 is oriented toward V . These edges are mapped by ϕ to A j X, j = 1, 2, 3, and we must have A 1 X → m XA 2 and A 1 X → m XA 3 . It follows that the edge XB opposite A 1 X satisfies m(XB) = 0, and therefore m 1 (XB) = 0, so that this vertex V contributes nothing to Σ m 2 (m 1 ).…”