Related papers: PBW degenerate Schubert varieties: Cartan componen…

We concern the VIGRE's conjecture; namely the complexity of a Specht module is the p-weight of the corresponding partition if and only if the partition is not p by p. In abelian defect case, we calculate the cohomological variety of the…

The goal of this paper is to study the link between the topology of the degenerate flag varieties and combinatorics of the Dellac configurations. We define three new classes of algebraic varieties closely related to the degenerate flag…

We discuss the deformed function algebra of a simply connected reductive Lie group G over the complex numbers using a basis consisting of matrix elements of finite dimensional representations. This leads to a preferred deformation, meaning…

A result of Zelevinsky states that an orbit closure in the space of representations of the equioriented quiver of type $A_h$ is in bijection with the opposite cell in a Schubert variety of a partial flag variety $SL(n)/Q$. We prove that…

We establish P=W and PI=WI conjectures for character varieties with structural group $\mathrm{GL}_n$ and $\mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W…

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…

We propose a theory of degenerations for derived module categories, analogous to degenerations in module varieties for module categories. In particular we define two types of degenerations, one algebraic and the other geometric. We show…

For a quotient algebra $U$ of the tensor algebra we give explicit conditions on its relations for $U$ being a PBW-deformation of an $N$-Koszul algebra $A$. We show there is a one-one correspondence between such deformations and a class of…

The introduction is modified in the revised version. Also, many typos and errors were corrected. Let $W\to C$ be degeneration of smooth varieties so that the special fiber has normal crossing singularity. In this paper, we first constructed…

We study the PBW filtration on irreducible finite--dimensional representations for the Lie algebra of type $\tt B_n$. We prove in several cases, including all multiples of the adjoint representation and all irreducible finite--dimensional…

Quiver Grassmannians are projective varieties parametrizing subrepresentations of given dimension in a quiver representation. We define a class of quiver Grassmannians generalizing those which realize degenerate flag varieties. We show that…

The Schubert varieties on a flag manifold G/P give rise to a cell decomposition on G/P whose Kronecker duals, known as the Schubert classes on G/P, form an additive base of the integral cohomology of G/P. The Schubert's problem of…

We construct a basis for a modified quantum group of finite type, extending the PBW bases of positive and negative halves of a quantum group. Generalizing Lusztig's classic results on PBW bases, we show that this basis is orthogonal with…

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…

For an untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ with Cartan and Borel subalgebras $\mathfrak{h} \subset \mathfrak{b} \subset \mathfrak{g}$, affine Demazure modules are certain $U(\mathfrak{b})$-submodules of the irreducible…

Let $G$ be a simply connected, almost simple group over an algebraically closed field $\mathbf k$, and $P$ a maximal parabolic subgroup corresponding to omitting a cominuscule root. We construct a compactification $\phi:T^*G/P\rightarrow…

We generalize the classification of isomorphism classes of Schubert varieties in complete flag varieties G/B to a class of partial flag varieties G/P. In particular, we classify all Schubert varieties in G/P where P is a minimal parabolic…

For $G$ a reductive group and $T\subset B$ a maximal torus and Borel subgroup, Demazure modules are certain $B$-submodules, indexed by elements of the Weyl group, of the finite irreducible representations of $G$. In order to describe the…

In this chapter, we give an overview of Billey-Postnikov (BP) decompositions which have become an important tool for understanding the geometry and combinatorics of Schubert varieties. BP decompositions are factorizations of Coxeter group…

Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the…