Flag arrangements and triangulations of products of simplices

We investigate the line arrangement that results from intersecting d complete flags in C^n. We give a combinatorial description of the matroid T_{n,d} that keeps track of the linear dependence relations among these lines. We prove that the bases of the matroid T_{n,3} characterize the triangles with holes which can be tiled with unit rhombi. More generally, we provide evidence for a conjectural connection between the matroid T_{n,d}, the triangulations of the product of simplices Delta_{n-1} x \Delta_{d-1}, and the arrangements of d tropical hyperplanes in tropical (n-1)-space. Our work provides a simple and effective criterion to ensure the vanishing of many Schubert structure constants in the flag manifold, and a new perspective on Billey and Vakil's method for computing the non-vanishing ones.Comment: 39 pages, 12 figures, best viewed in colo

Lower bounds for Kazhdan-Lusztig polynomials from patterns

We give a lower bound for the value at q=1 of a Kazhdan-Lustig polynomial in a Weyl group W in terms of "patterns''. This is expressed by a "pattern map" from W to W' for any parabloic subgroup W'. This notion generalizes the concept of patterns and pattern avoidance for permutations to all Weyl groups. The main tool of the proof is a "hyperbolic localization" on intersection cohomology; see the related paper http://front.math.ucdavis.edu/math.AG/0202251Comment: 14 pages; updated references. Final version; will appear in Transformation Groups vol.8, no.

Fingerprint databases for theorems

We discuss the advantages of searchable, collaborative, language-independent databases of mathematical results, indexed by "fingerprints" of small and canonical data. Our motivating example is Neil Sloane's massively influential On-Line Encyclopedia of Integer Sequences. We hope to encourage the greater mathematical community to search for the appropriate fingerprints within each discipline, and to compile fingerprint databases of results wherever possible. The benefits of these databases are broad - advancing the state of knowledge, enhancing experimental mathematics, enabling researchers to discover unexpected connections between areas, and even improving the refereeing process for journal publication.Comment: to appear in Notices of the AM

A vector partition function for the multiplicities of sl_k(C)

We use Gelfand-Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra sl_k(C) (type A_{k-1}) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to that of the Duistermaat-Heckman measure from symplectic geometry, which gives a large-scale limit way to look at multiplicity diagrams. We also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors. Using symplectic geometry, we prove that the partition of the permutahedron into domains of polynomiality of the Duistermaat-Heckman function is the same as that for the weight multiplicity function, and give an elementary proof of this for sl_4(C) (A_3).Comment: 34 pages, 11 figures and diagrams; submitted to Journal of Algebr