Related papers: Generalized juggling patterns, quiver Grassmannian…
In this paper, a description of the set-theoretical defining equations of symplectic (type C) Grassmannian/flag/Schubert varieties in corresponding (type A) algebraic varieties is given as linear polynomials in Pl$\ddot{u}$cker coordinates,…
We study torus-equivariant algebraic $K$-theory of affine Schubert varieties in the perfect affine Grassmannians over $\mathbb{F}_p$. We further compare it to the torus-equivariant Hochschild homology of perfect complexes, which has a…
This review is dedicated to two-dimensional sigma models with flag manifold target spaces, which are generalizations of the familiar $CP^{n-1}$ and Grassmannian models. They naturally arise in the description of continuum limits of spin…
The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroids and plabic graphs. Remarkably, the same combinatorial…
We study fixed-point loci of Nakajima varieties under symplectomorphisms and their anti-symplectic cousins, which are compositions of a diagram automorphism, a reflection functor and a transpose defined by certain bilinear forms. These…
We prove a short, root-system uniform, combinatorial classification of Levi-spherical Schubert varieties for any generalized flag variety $G/B$ of finite Lie type. We apply this to the study of multiplicity-free decompositions of a Demazure…
In this paper, we provide a new construction of quiver algebroid stacks and the associated mirror functors for symplectic manifolds. First, we formulate the concept of a quiver stack, which is a geometric structure formed by gluing multiple…
This paper determines the relationship between the geometry of retractions and the combinatorics of folded galleries for arbitrary affine buildings, and so provides a unified framework to study orbits in affine flag varieties. We introduce…
The Zariski closures of the orbits for representations of type A Dynkin quivers under the action of general linear groups (i.e. quiver loci) exhibit a profound connection with Schubert varieties. In this paper, we present a…
We construct new half-BPS line defects in 3d $\mathcal{N}=2$ supersymmetric quiver gauge theories whose Higgs branches are complete flag manifolds $X = {\rm Fl}(n)$. Upon circle compactification, the bulk theory flows to a non-linear sigma…
We study the representation theory of quantizations of Gieseker moduli spaces. Namely, we prove the localization theorems for these algebras, describe their finite dimensional representations and two-sided ideals as well as their categories…
In a previous paper we defined the concept of an affinized projective variety and its associated Hilbert series. We computed the Hilbert series for varieties associated to quadratic monomial ideals. In this paper we show how to apply these…
We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type $A$. We introduce Murnaghan-Nakayama elements and Dunkl elements in the…
We prove a criterion for the normality of Schubert varieties in twisted affine Grassmannians in terms of the order of the algebraic fundamental group of a certain Levi subgroup, in particular in small positive characteristic. As an…
We study quiver Grassmannians associated with indecomposable representations of the Kronecker quiver. We find a cellular decomposition of them and we compute their Betti numbers. As an application, we give a geometric realization of the…
In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear…
We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. Along the way, we give new proofs of the presentation of the…
Let I be a finite set and CI be the algebra of functions on I. For a finite dimensional C algebra A with \CI contained in A we show that certain moduli spaces of finite dimsional modules are isomorphic to certain Grassmannian (quot-type)…
We propose a combinatorial model for the Schubert structure constants of the complete flag manifold when one of the factors is Grassmannian.
In this paper, we introduce morphisms for matroids with coefficients (in the sense of Baker and Bowler) and quiver matroids. We investigate their basic properties, such as functoriality, duality, minors and cryptomorphic characterizations…