相关论文: Newton-Okounkov polytopes of flag varieties for cl…

A generic polynomial f(x,y,z) with a prescribed Newton polytope defines a symmetric spatial curve f(x,y,z)=f(y,x,z)=0. We study its geometry: the number, degree and genus of its irreducible components, the number and type of singularities,…

Motivated by the construction of Newton--Okounkov bodies and toric degenerations via cluster algebras in [GHKK18, FO25], we consider a family of Newton--Okounkov polytopes of a complex smooth Fano variety $X$ related by a composition of…

Let $G$ be a classical group defined over the complex numbers with a Borel subgroup $B$. Choose a holomorphic involution of $G$ and let $K$ be its set of fixed points. The group $K$ acts on the flag variety $G/B$ with finitely many orbits…

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl_2 and sl_3 and by Mazorchuk-Stroppel…

We give the construction of weighted Lagranngiann Grassmannians$wLGr(3,6)$ and weighted partial $A_3$ flag variety $wFl_{1,3}$ coming from the symplectic Lie group $Sp(6,\mathbb C)$ and the general linear group $GL(4,\mathbb C)$…

We construct Koppelman formulas on manifolds of flags in $\C^N$ for forms with values in any holomorphic line bundle as well as in the tautological vector bundles and their duals. As an application we obtain new explicit proofs of some…

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…

The full root polytope of type $A$ is the convex hull of all pairwise differences of the standard basis vectors which we represent by forward and backward arrows. We completely classify all flag triangulations of this polytope that are…

We show that canonical bases in $\dot{U}(\mathfrak{sl}_n)$ and the Schur algebra are compatible; in fact we extend this result to $p$-canonical bases. This follows immediately from a fullness result from a functor categorifying this map. In…

Based on recent advances on the relation between geometry and representation theory, we propose a new approach to elliptic Schubert calculus. We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized…

Abstract polytopes generalize the classical notion of convex polytopes to more general combinatorial structures. The most studied ones are regular and chiral polytopes, as it is well-known, they can be constructed as coset geometries from…

Classifications of $\rm{SL}(n)$ covariant function-valued valuations are established with some assumptions of continuity. New valuations, for example, weighted moment functions, are introduced and our classifications give unified…

Let $Y$ be a (partial) minimal model of a scheme $V$ with a cluster structure. Under natural assumptions, for every choice of seed we associate a Newton--Okounkov body to every divisor on $Y$ supported on $Y \setminus V$ and show that these…

Wythoff's construction associates a uniform polytope to a Coxeter diagram whose vertices are decorated with crosses, which indicate the subgroup stabilizing a generic point. Champagne, Kjiri, Patera, and Sharp remarked that by associating…

The paper deals with three topics on coquasitriangular bialgebras. A characterization of universal r-forms in terms of Yetter-Drinfeld modules is given. All universal r-forms for the coordinate Hopf algebras of the quantum groups GL_q(N),…

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $K$-theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes…

We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's…

We generalize the theory of Newton-Okounkov bodies of big divisors to the case of graded linear series. One of the results is the generalization of slice formulas and the existence of generic Newton-Okounkov bodies for birational graded…

In this paper, we study the Gelfand--Cetlin systems and polytopes of the co-adjoint $\mathrm{SO}(n)$-orbits. We describe the face structure of Gelfand--Cetlin polytopes and iterated bundle structure of Gelfand--Cetlin fibers in terms of…

For N=5, 6 and 7, using the classification of perfect quadratic forms, we compute the homology of the Voronoi cell complexes attached to the modular groups SL_N(\Z) and GL_N(\Z). From this we deduce the rational cohomology of those groups.