scholarly journals Cluster Structures on Double Bott–Samelson Cells

2021 ◽  
Vol 9 ◽  
Author(s):  
Linhui Shen ◽  
Daping Weng
Keyword(s):  
Moduli Spaces ◽  
Rational Functions ◽  
Cartan Matrix ◽  
Cluster Algebras ◽  
Geometric Proof ◽  
Link Invariants ◽  
Constructible Sheaves ◽  
Legendrian Links ◽  
Smooth Affine ◽  
Coordinate Rings

Abstract Let $\mathsf {C}$ be a symmetrisable generalised Cartan matrix. We introduce four different versions of double Bott–Samelson cells for every pair of positive braids in the generalised braid group associated to $\mathsf {C}$ . We prove that the decorated double Bott–Samelson cells are smooth affine varieties, whose coordinate rings are naturally isomorphic to upper cluster algebras. We explicitly describe the Donaldson–Thomas transformations on double Bott–Samelson cells and prove that they are cluster transformations. As an application, we complete the proof of the Fock–Goncharov duality conjecture in these cases. We discover a periodicity phenomenon of the Donaldson–Thomas transformations on a family of double Bott–Samelson cells. We give a (rather simple) geometric proof of Zamolodchikov’s periodicity conjecture in the cases of $\Delta \square \mathrm {A}_r$ . When $\mathsf {C}$ is of type $\mathrm {A}$ , the double Bott–Samelson cells are isomorphic to Shende–Treumann–Zaslow’s moduli spaces of microlocal rank-1 constructible sheaves associated to Legendrian links. By counting their $\mathbb {F}_q$ -points we obtain rational functions that are Legendrian link invariants.

PL Link Isotopy, Essential Knotting and Quotients of Polynomials

1991 ◽  
Vol 34 (4) ◽  
pp. 536-541 ◽  
Author(s):  
Dale Rolfsen
Keyword(s):  
Knot Theory ◽  
Piecewise Linear ◽  
Rational Functions ◽  
Isotopy Class ◽  
Link Invariants ◽  
Link Theory ◽  
The Individual

AbstractPiecewise-linear (nonambient) isotopy of classical links may be regarded as link theory modulo knot theory. This note considers an adaptation of new (and old) polynomial link invariants to this theory, obtained simply by dividing a link's polynomial by the polynomials of the individual components. The resulting rational functions are effective in distinguishing isotopy classes of links, and in demonstrating that certain links are essentially knotted in the sense that every link in its isotopy class has a knotted component. We also establish geometric criteria for essential knotting of links.

Endomorphisms of Kronecker Modules Regulated by Quadratic Algebra Extensions of a Function Field

2008 ◽  
Vol 60 (4) ◽  
pp. 923-957 ◽  
Author(s):  
F. Okoh ◽  
F. Zorzitto
Keyword(s):  
Height Function ◽  
Rational Functions ◽  
Trivial Extension ◽  
Closed Field ◽  
Quadratic Algebra ◽  
Endomorphism Algebra ◽  
Finite Dimensional ◽  
Free Rank ◽  
Rank 2 ◽  
Coordinate Rings

AbstractThe Kronecker modules , where m is a positive integer, h is a height function, and α is a K-linear functional on the space K(X) of rational functions in one variable X over an algebraically closed field K, aremodels for the family of all torsion-free rank-2 modules that are extensions of finite-dimensional rank-1 modules. Every such module comes with a regulating polynomial f in K(X)[Y]. When the endomorphism algebra of is commutative and non-trivial, the regulator f must be quadratic in Y. If f has one repeated root in K(X), the endomorphismalgebra is the trivial extension for some vector space S. If f has distinct roots in K(X), then the endomorphisms forma structure that we call a bridge. These include the coordinate rings of some curves. Regardless of the number of roots in the regulator, those End that are domains have zero radical. In addition, each semi-local End must be either a trivial extension or the product K × K.

scholarly journals Positivity in Coefficient-Free Rank Two Cluster Algebras

10.37236/187 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
G. Dupont
Keyword(s):  
Short Note ◽  
Rational Functions ◽  
Laurent Polynomial ◽  
Cluster Algebras ◽  
Free Rank ◽  
Positive Integers

Let $b,c$ be positive integers, $x_1,x_2$ be indeterminates over ${\Bbb Z}$ and $x_m, m \in {\Bbb Z}$ be rational functions defined by $x_{m-1}x_{m+1}=x_m^b+1$ if $m$ is odd and $x_{m-1}x_{m+1}=x_m^c+1$ if $m$ is even. In this short note, we prove that for any $m,k \in {\Bbb Z}$, $x_k$ can be expressed as a substraction-free Laurent polynomial in ${\Bbb Z}[x_m^{\pm 1},x_{m+1}^{\pm 1}]$. This proves Fomin-Zelevinsky's positivity conjecture for coefficient-free rank two cluster algebras.

scholarly journals On $$\mathsf {G} $$-isoshtukas over function fields

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Paul Hamacher ◽  
Wansu Kim
Keyword(s):  
Moduli Spaces ◽  
Function Fields ◽  
Rational Functions ◽  
Conjugacy Classes ◽  
Projective Curve ◽  
Smooth Projective Curve

AbstractIn this paper we classify isogeny classes of global $$\mathsf {G} $$ G -shtukas over a smooth projective curve $$C/{\mathbb {F}}_q$$ C / F q (or equivalently $$\sigma $$ σ -conjugacy classes in $$\mathsf {G} (\mathsf {F} \otimes _{{\mathbb {F}}_q} \overline{{\mathbb {F}}_q})$$ G ( F ⊗ F q F q ¯ ) where $$\mathsf {F} $$ F is the field of rational functions of C) by two invariants $${\bar{\kappa }},{\bar{\nu }}$$ κ ¯ , ν ¯ extending previous works of Kottwitz. This result can be applied to study points of moduli spaces of $$\mathsf {G} $$ G -shtukas and thus is helpful to calculate their cohomology.

scholarly journals Affine cluster monomials are generalized minors

2019 ◽  
Vol 155 (7) ◽  
pp. 1301-1326
Author(s):  
Dylan Rupel ◽  
Salvatore Stella ◽  
Harold Williams
Keyword(s):  
Lie Groups ◽  
Cluster Algebras ◽  
Quiver Representations ◽  
Group Representations ◽  
Finite Dimensional ◽  
Bruhat Cells ◽  
Level Zero ◽  
Coordinate Rings ◽  
Affine Type

We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with$\mathbf{g}$-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type$A_{1}^{(1)}$, we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.

scholarly journals Bott–Samelson Varieties and Poisson Ore Extensions

2019 ◽  
Author(s):  
Balázs Elek ◽  
Jiang-Hua Lu
Keyword(s):  
Total Positivity ◽  
Cluster Algebras ◽  
Poisson Brackets ◽  
Ore Extension ◽  
Bruhat Cells ◽  
Toric Degenerations ◽  
Ore Extensions ◽  
Structure Constants ◽  
Poisson Varieties ◽  
Coordinate Rings

Abstract We show that associated with any $n$-dimensional Bott–Samelson variety of a complex semi-simple Lie group $G$, one has $2^n$ Poisson brackets on the polynomial algebra $A={\mathbb{C}}[z_1, \ldots , z_n]$, each an iterated Poisson Ore extension and one of them a symmetric Poisson Cauchon–Goodearl–Letzter (CGL) extension in the sense of Goodearl–Yakimov. We express the Poisson brackets in terms of root strings and structure constants of the Lie algebra of $G$. It follows that the coordinate rings of all generalized Bruhat cells have presentations as symmetric Poisson CGL extensions. The paper establishes the foundation on generalized Bruhat cells and sets the stage for their applications to integrable systems, cluster algebras, total positivity, and toric degenerations of Poisson varieties, some of which are discussed in the Introduction.

The Convex Polytopes and Homogeneous Coordinate Rings of Bivariate Polynomials

2019 ◽  
Vol 16 (2) ◽  
pp. 1
Author(s):  
Shamsatun Nahar Ahmad ◽  
Nor’Aini Aris ◽  
Azlina Jumadi
Keyword(s):  
Convex Polytopes ◽  
Polynomial Systems ◽  
Matrix Formulation ◽  
Bivariate Polynomials ◽  
Homogeneous Coordinates ◽  
The Matrix ◽  
Projective Vector ◽  
Coordinate Rings ◽  
Resultant Matrix ◽  
The Given

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

Sign in / Sign up

Export Citation Format

Share Document