Polynomial identities related to special Schubert varieties

Let $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

Poincaré polynomials of generic torus orbit closures in Schubert varieties

The closure of a generic torus orbit in the flag variety G / B G/B of type A A is known to be a permutohedral variety, and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in G / B G/B . When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.

The BdR+-affine Grassmannian

This chapter defines an object that was one of the big motivations to develop a theory of diamonds. In the study of the usual Grassmannian variety G/B attached to a reductive group G, one defines a Schubert variety to be the closure of a B-orbit in G/B. Generally, Schubert varieties are singular varieties. Desingularizations of Schubert varieties are constructed by Demazure. The chapter uses an analogue of this construction in the context of the B+ dR-Grassmannian. It then looks at miniscule Schubert varieties. In this case, one can identify the space explicitly. If µ is minuscule, the Bialynicki–Birula map is an isomorphism.

Toric matrix Schubert varieties and root polytopes (extended abstract)

International audience Start with a permutation matrix π and consider all matrices that can be obtained from π by taking downward row operations and rightward column operations; the closure of this set gives the matrix Schubert variety Xπ. We characterize when the ideal defining Xπ is toric (with respect to a 2n − 1-dimensional torus) and study the associated polytope of its projectivization. We construct regular triangulations of these polytopes which we show are geometric realizations of a family of subword complexes. We also show that these complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of β-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We also write the volume and Ehrhart series of root polytopes in terms of β-Grothendieck polynomials. Subword complexes were introduced by Knutson and Miller in 2004, who showed that they are homeomorphic to balls or spheres and raised the question of their polytopal realizations.

Explicit decomposition theorem for special Schubert varieties

We provide an explicit canonical description of the perverse cohomology sheaves and of the primitive perverse cohomology complexes for the non-small resolution {\pi:\tilde{\mathcal{S}}\to\mathcal{S}} of a special Schubert variety {\mathcal{S}}. For such a resolution, we also discuss a way to obtain an explicit splitting of {R\pi{{}_{*}}\mathbb{Q}_{\widetilde{{\mathcal{S}}}}}, in the derived category, by means of Gysin morphisms and cohomology extensions.

Diagrams and Essential Sets for Signed Permutations

We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations. In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation. Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.

Frobenius Splitting of Thick Flag Manifolds of Kac–Moody Algebras

We explain that the Plücker relations provide the defining equations of the thick flag manifold associated to a Kac–Moody algebra. This naturally transplants the result of Kumar–Mathieu–Schwede about the Frobenius splitting of thin flag varieties to the thick case. As a consequence, we provide a description of the space of global sections of a line bundle of a thick Schubert variety as conjectured in Kashiwara–Shimozono [13]. This also yields the existence of a compatible basis of thick Demazure modules and the projective normality of the thick Schubert varieties.