Related papers: Pattern characterization of rationally smooth affi…

The aim of this article is to present a smoothness criterion for Schubert varieties in generalized flag manifolds $G/B$ in terms of patterns in root systems. We generalize Lakshmibai-Sandhya's well-known result that says that a Schubert…

Let G be a simply-connected simple compact Lie group over the complex numbers. The affine Grassmannian is a projective ind-variety, homotopy-equivalent to the loop space of G and closely analogous to a maximal flag variety of the classical…

We show that every smooth Schubert variety of affine type $\tilde{A}$ is an iterated fibre bundle of Grassmannians, extending an analogous result by Ryan and Wolper for Schubert varieties of finite type $A$. As a consequence, we finish a…

Let $P$ be a parabolic subgroup in $G=SL_n(\mathbf k)$, for $\mathbf k$ an algebraically closed field. We show that there is a $G$-stable closed subvariety of an affine Schubert variety in an affine partial flag variety which is a natural…

We give a permutation pattern avoidance criteria for determining when the projection map from the flag variety to a Grassmannian induces a fiber bundle structure on a Schubert variety. In particular, we introduce the notion of a split…

We study the toric degeneration of Weyl group translated Schubert divisors of a partial flag variety of Lie type A via Gelfand-Cetlin polytopes. We propose a conjecture that Schubert varieties of appropriate dimensions intersect…

We study the geometry of equicharacteristic partial affine flag varieties associated to tamely ramified groups $G$ in characteristics $p>0$ dividing the order of the fundamental group $\pi_1(G_{\text{der}})$. We obtain that most Schubert…

We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of *interval pattern avoidance*. For "reasonable" invariants…

Schubert varieties in the full flag variety of Kac-Moody type are indexed by elements of the corresponding Weyl group. We give a practical criterion for when two such Schubert varieties (from potentially different flag varieties) are…

We definitively establish that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on…

We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field $\mathbb K$ of characteristic $\neq 2$ from scratch. We show that the formal model…

Schubert varieties are irreducible subvarieties of homogeneous manifold, which are important to understand the geometry of homogeneous manifold G/P and the action of the semisimple Lie group G. Consider the space of effective cycles in G/P…

We provide a proof that every Schubert variety of a semi-infinite flag variety is projectively normal. This gives us an interpretation of a Demazure module of a global Weyl module of a current Lie algebra as the (dual) space of the space of…

We describe the loci of non-rationally smooth (nrs) points and of singular points for any non-spiral Schubert variety of $\tilde{A}_2$ in terms of the geometry of the (affine) Weyl group action on the plane $\mathbb{R}^2$. Together with the…

We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the…

The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of…

We characterize Cohen--Macaulay and $\varphi$-rational perfect schemes in terms of their perverse \'etale mod $p$ sheaves. Using inversion of adjunction, we prove that sufficiently small Schubert varieties in the Witt affine flag variety…

Let G be a complex semi-simple linear algebraic group without G_2 factors, B a Borel subgroup of G and T a maximal torus in B. The flag variety G/B is a projective G-homogeneous variety whose tangent space at the identity coset is…

Flag domains are open orbits of real semisimple Lie groups in flag manifolds of their complexifications. Certain group theoretically defined compact complex submanifolds, which are regarded as cycles, are of basic importance for their…

In this article, we investigate the toric Schubert varieties in partial flag varieties $G/P$ for a connected semisimple algebraic group $G$. Using Deodhar's decomposition of Richardson varieties and the work of Pasquier, we give an explicit…