The Space of Triangles, Vanishing Theorems, and Combinatorics

Abstract

We consider compactifications of P R j ⌬ , the space of triples of distinct i j points in projective space. One such space is a singular variety of configurations of points and lines; another is the smooth compactification of Fulton and MacPherson; and a third is the triangle space of Schubert and Semple.We compute the sections of line bundles on these spaces, and show that they are Ž . equal as GL n representations to the generalized Schur modules associated to Ž . ''bad'' generalized Young diagrams with three rows Borel᎐Weil theorem . On the one hand, this yields Weyl-type character and dimension formulas for the Schur modules; on the other, a combinatorial picture of the space of sections. Cohomology vanishing theorems play a key role in our analysis.

A Smooth Space of Tetrahedra

“…The difference between Schubert's space and Fulton-MacPherson-Roberts-Speiser's space appears when one considers the torus action on them, cf [18]. …”

“…In [14], Schubert described a desingularization of X, and used it to study enumerative problems involving triangles [15,13,3]. • The space X is a configuration variety in the sense of Magyar [10,18]. Such spaces arise naturally in the study of generalized Schur modules.…”

Perverse schobers and birational geometry

“…We recall [34,45,47] the definition of T Sch going back to Schubert [46]. Consider the 6-dimensional space S 2 (k 3 * ) of quadratic forms on k 3 .…”

A representation-theoretic interpretation of positroid classes

“…In this section we observe that every three-row diagram D is (essentially) a toric skew shape, so that the results of Sections 3 and 7.1 give a combinatorial algorithm for decomposing V [D] into irreducibles. For a different approach, see [vdKM99], which describes an algorithm for writing ch V [D] as a sum of rational functions whenever D is a three-row diagram.…”