Eulerian numbers, tableaux, and the Betti numbers of a toric variety

Abstract

Let [Sigma] denote the Coxeter complex of Sn, and let X([Sigma]) denote the associated toric variety. Since the Betti numbers of the cohomology of X([Sigma]) are Eulerian numbers, the additional presence of an Sn-module structure permits the definition of an isotypic refinement of these numbers. In some unpublished work, DeConcini and Procesi derived a recurrence for the Sn-character of the cohomology of X([Sigma]), and Stanley later used this to translate the problem of combinatorially describing the isotypic Eulerian numbers into the language of symmetric functions. In this paper, we explicitly solve this problem by developing a new way to use marked sequences to encode permutations. This encoding also provides a transparent explanation of the unimodality of Eulerian numbers and their isotypic refinements.