Related papers: Canonical bases and quiver varieties

We show that Lusztig's canonical basis for the degree two part of the Grassmannian coordinate ring is given by SL(k) web diagrams. Equivalently, we show that every SL(2) web immanant of a plabic graph for Gr(k,n) is an SL(k) web invariant.

Let ${\mathbf U}_q^-$ be the negative half of a quantum group of finite type. We construct the canonical basis of ${\mathbf U}_q^-$ by applying the folding theory of quantum groups, and piecewise linear parametrization of canonical basis.…

We generalize a construction in [BW18] (arXiv:1610.09271) by showing that the tensor product of a based $\textbf{U}^{\imath}$-module and a based $\textbf{U}$-module is a based $\textbf{U}^{\imath}$-module. This is then used to formulate a…

We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. As a by-product, we give a counterexample…

Some filtrations of the tensor product of a highest weight module and a lowest weight module over quantum group $U_q(\mathfrak g)$ are constructed in \cite{LZ:2009} and one can use them to define some ideals of the modified quantized…

Berenstein and Zelevinsky introduced quantum cluster algebras [Adv. Math, 2005] and the triangular bases [IMRN, 2014]. The support conjecture by Lee-Li-Rupel-Zelevinsky [PNAS, 2014] asserts that the support of a triangular basis element for…

We propose a construction of some canonical bases for quantum loop algebras of Kac-Moody algebras. We consider a smooth projective curve X, a group of automorphism G of X such that X/G=P^1, and we consider some Quot schemes of G-equivariant…

In previous work, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral…

We study classes determined by the Kazhdan-Lusztig basis of the Hecke algebra in the $K$-theory and hyperbolic cohomology theory of flag varieties. We first show that, in $K$-theory, the two different choices of Kazhdan-Lusztig bases…

We express notions of K-stability of polarized spherical varieties in terms of combinatorial data, vastly generalizing the case of toric varieties. We then provide a combinatorial sufficient condition of G-uniform K-stability by studying…

We suggest a (conjectural) construction of a basis in the plus part of the affine Lie algebra of type ADE indexed by irreducible components of certain quiver varieties. This construction is closely related to a string-theoretic construction…

We describe a natural $q$-deformation of Fock and Goncharov's canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type $A$. We then describe an extension of this construction involving a…

We show the existence of canonical heights of subvarieties for bounded sequences of morphisms and give some applications.

We identify the canonical basis of the quantum adjoint representation of a quantized enveloping algebra with a basis that we defined before the theory of canonical bases was available.

We study preprojective algebras of graphs and their relationship to module categories over representations of quantum SL(2). As an application, ADE quiver varieties of Nakajima are shown to be subvarieties of the variety of representations…

We prove an asymptotic formula conjectured by Manin for the number of $K$-rational points of bounded height with respect to the anticanonical line bundle for arbitrary smooth projective toric varieties over a number field $K$.

We obtain a family of explicit "polyhedral" combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra. Here "polyhedral" means that the multiplicity in…

Shafarevich's hyperbolicity conjecture asserts that a family of curves over a quasi-projective 1-dimensional base is isotrivial unless the logarithmic Kodaira dimension of the base is positive. More generally it has been conjectured by…

We describe the equivariant K-groups of a family of generalized Steinberg varieties that interpolates between the Steinberg variety of a reductive, complex algebraic group and its nilpotent cone in terms of the extended affine Hecke algebra…

We give a combinatorial construction for the canonical bases of the $\pm$-parts of the quantum enveloping superalgebra $\bfU(\mathfrak{gl}_{m|n})$ and discuss their relationship with the Kazhdan-Lusztig bases for the quantum Schur…