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Regular semisimple Hessenberg varieties are subvarieties of the flag variety $\mathrm{Flag}(\mathbb{C}^n)$ arising naturally in the intersection of geometry, representation theory, and combinatorics. Recent results of…
We study certain mild degenerations of algebraic varieties which appear in the analysis of a large class of supersymmetric theories, including superstring theory. We analyze Witten's sigma-model and find that the non-transversality of the…
The ring K(G/B) is isomorphic to a quotient of a Laurent polynomial ring by an ideal generated by certain W-symmetric functions and has a basis given by classes O_w, where O_w is the class of the structure sheaf of the Schubert variety X_w.…
We explain that the Pl\"ucker relations provide the defining equations of the thick flag manifold associated to a Kac-Moody algebra. This naturally transplant the result of Kumar-Mathieu-Schwede about the Frobenius splitting of thin flag…
The coherence conjecture of Pappas and Rapoport, proved by Zhu, asserts the equality of dimensions for the global sections of a line bundle over a spherical Schubert variety in the affine Grassmannian and those of another line bundle over a…
We study aspects of superstring vacua of non-compact special holonomy manifolds with conical singularities constructed systematically using soluble N = 1 superconformal field theories (SCFT's). It is known that Einstein homogeneous spaces…
We study the relation between the kappa-symmetric formulation of the supermembrane in eleven dimensions and the pure-spinor version. Recently, Berkovits related the Green-Schwarz and pure-spinor superstrings. In this paper, we attempt to…
We describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of faces of polytopes. For GL_n and Gelfand--Zetlin polytopes, combinatorics of this algorithm coincides with that of…
We give an explicit subword complex description of the generators of the type cone of the g-vector fan of a finite type cluster algebra with acyclic initial seed. This yields in particular a description of the Newton polytopes of the…
We use the connective formal group law to define a one-parameter ($\beta$-)deformation of the motivic Segre classes of Schubert cells in the $d$-step flag variety. This $\beta$-deformation specializes to the motivic Segre classes of…
We give a description of faces of all codimensions for the cones of weights of rings of semi-invariants of quivers. For a triple flag quiver and faces of codimension 1 this reduces to the result of Knutson-Tao-Woodward on the facets of the…
We describe the torus-equivariant cohomology of weighted partial flag orbifolds ${\mathrm{w}}\Sigma$ of type $A$. We establish counterparts of several results known for the partial flag variety that collectively constitute what we refer to…
We define a very general class of CHL-models associated with any string theory (bosonic or supersymmetric) compactified on an internal CFT C x T^d. We take the orbifold by a pair (g,\delta), where g is a (possibly non-geometric) symmetry of…
We introduce rectangular elements in the symmetric group. In the framework of PBW degenerations, we show that in type A the degenerate Schubert variety associated to a rectangular element is indeed a Schubert variety in a partial flag…
We establish an explicit combinatorial/homological characterization of supports for linear degenerations of flag varieties. For such purpose, we introduce the concept of an excessive multisegment. It provides a new class of combinatorial…
One approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Through an identification of the cohomology ring of the type A full flag variety with the polytope ring of the…
This paper discusses some results on semiclassical string configurations lying in the IR sector of supergravity backgrounds dual to confining gauge theories. On the gauge theory side, the string states we analyse correspond to Wilson loops,…
We show that in type A or C any degenerate flag variety is in fact isomorphic to a Schubert variety in an appropriate partial flag manifold.
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…
Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator $X$ and a nondecreasing function $h$. The family of Hessenberg varieties for regular $X$ is particularly important: they are used in quantum…