Searcharxiv⌕ Search

arXiv subjects

Guanglian Zhang

Publications and source records attributed to Guanglian Zhang.

9 recordsLinked to original sources
arXiv2024

Every type A quiver locus is a Kazhdan-Lusztig variety

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 scheme-theoretical isomorphism between a type A quiver locus and the intersection of an opposite Schubert cell and a Schubert variety, also known as a Kazhdan-Lusztig variety in geometric representation theory. Our results generalize and unify the Zelevinsky maps for equioriented type A quiver loci and bipartite type A quiver loci, as presented respectively by A. V. Zelevinsky in 1985 and by R. Kinser and J. Rajchgot in 2015. Through this isomorphism, we establish a direct and natural connection between type A quiver loci and Schubert varieties. We compute an explicit relationship between Zelevinsky permutations and the indecomposable factors of the corresponding representations. Additionally, we present three applications of our isomorphism with examples in order to justify its further potentials.

math.AG↗
arXiv2023

Symplectic conditions on Grassmannian, flag, and Schubert varieties

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, and it is proved that such equations generate the defining ideal of variety of type C in those of type A. As applications of this result, the number of local equations required to obtain the Schubert variety of type C from the Schubert variety of type A is computed, and further geometric properties of the Schubert variety of type C are given in the aspect of complete intersections. Finally, the smoothness of Schubert variety in the non-minuscule or cominuscule Grassmannian of type C is discussed, filling gaps in the study of algebraic varieties of the same type.

math.AG↗
arXiv2020

The defining equations of a class of Richardson and flag varieties on Sp$_{2n}(k)$

This paper aims to focus on Richardson varieties on symplectic groups, especially their combinatorial characterization and defining equations. Schubert varieties and opposite Schubert varieties have profound significance in the study of generalized flag varieties which are not only research objects in algebraic geometry but also ones in representation theory. A more general research object is Richardson variety, which is obtained by the intersection of a Schubert variety and an opposite Schubert variety. The structure of Richardson variety on Grassmannian and its combinatorial characterization are well known, and there are also similar method on quotients of symplectic groups. In the first part of this paper, we calculate the orbit of the symplectic group action, and then rigorously give a method to describe the corresponding quotient by using the nesting subspace sequence of the linear space, i.e. flags. At the same time, the flag is used to describe the Schubert variety and Richardson variety on quotient of symplectic group. The flag varieties of Sp_{2n}(k)/P_d can be viewed as closed subvarieties of Grassmannian. Using the standard monomial theory, we obtain the generators of its ideal, i.e. its defining equations, in homogeneous coordinate ring of Grassmannian. Furthermore, we prove several properties of the type C standard monomial on the symplectic group flag variety. Defining equations of Richardson varieties on Sp_{2n}(k)/P_d are given as well.

math.AG↗
arXiv2017

Rational Ringel-Hall algebras, Hall polynomials of affine type and Canonical bases

In this paper, the rational Ringel-Hall algebras for tame quivers are introduced and are identified with the positive part of the quantum extended Kac-Moody algebras. By using the rational Ringel-Hall algebras, we show that the existence of Hall polynomials for tame quiver algebras. The PBW bases are constructed and new classes of perverse sheaves are shown to have strong purity property. These allows us to construct the canonical bases of the positive part of the quantum extended Kac-Moody algebras.

math.RT↗
arXiv2016

Remark on a theorem in Mumford's Red Book of Varieties and Schemes

In this paper, we firstly point out, by a counter example, that Proposition 6.4 of Section 6 in Bump's book (Algebraic Geometry) is error, and then give a correct statement with proof. We finally point out a gap in the proof of Theorem 3, in Chapter I Section 8, of Mumford's red book, and indicate a way to complete it.

math.AG↗
arXiv2015

Canonical bases for the quantum extended Kac-Moody algebras and Hall polynomials

In this paper, the singular Ringel-Hall algebra for a tame quiver is introduced and shown to be isomorphic to the positive part of the quantum extended Kac-Moody algebra. A PBW basis is constructed and a new class of perverse sheaves is shown to have purity property. This allows to construct the canonical bases of the positive part of the quantum extended Kac-Moody algebra. As an application, the existence of Hall polynomials for tame quiver algebras is proved.

math.RT↗
arXiv2010

Derived Categories and Lie Algebras

Let $\md^b(A)$ be the derived category of a finite dimensional basic algebra $A$ with finite global dimension. We construct the Lie algebra arising from the 2-periodic version $\mk_2(\mp(A))$ of $\mk^b(\mp(A))$ in term of constructible functions on varieties attached to $\mk_2(\mp(A))$.

math.QA↗
arXiv2007

Representations of tame quivers and affine canonical bases

An integral PBW-basis of type $A_1^{(1)}$ has been constructed by Zhang [Z] and Chen [C] using the Auslander-Reiten quiver of the Kronecker quiver. We associate a geometric order to elements in this basis following an idea of Lusztig [L1] in the case of finite type. This leads to an algebraic realization of a bar-invariant basis of $\uq2$. For any affine symmetric type, we obtain an integral PBW-basis of the generic composition algebra, by using an algebraic construction of the integral basis for a tube in [DDX], an embedding of the module category of the Kronecker quiver into the module category of the tame quiver, and a list of the root vectors of indecomposable modules according to the preprojective, regular, and preinjective components of the Auslander-Reiten quiver of the tame quiver. When the basis elements are ordered to be compatible with the geometric order given by the dimensions of the orbit varieties and the extension varieties, we can show that the transition matrix between the PBW-basis and a monomial basis is triangular with diagonal entries equal to 1. Therefore we obtain a bar-invariant basis. By a orthogonalization for the PBW-basis with the inner product, we finally give an algebraic way to realize the canonical bases of the quantized enveloping algebras of all symmetric affine Kac-Moody Lie algebras.

math.QA↗