Related papers: Braid variety cluster structures, II: general type

We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n $\ge$ 3 strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov…

We give positive descriptions for certain Schubert structure constants $c_{u,v}^w$ for the full flag variety in Lie types $C$ and $D$. This is accomplished by first observing that a number of the $K=GL(n,\C)$-orbit closures on these flag…

We define an action of the extended affine d-strand braid group on the open positroid stratum in the Grassmannian Gr(k,n), for d the greatest common divisor of k and n. The action is by quasi-automorphisms of the cluster structure on the…

Given a knot or link in the handlebody, $H_g$, of genus $g$ we prove that it can always be represented as the plat closure of a braid in $H_g$. We further establish the Hilden braid group for the handlebody, as a subgroup of the mixed braid…

We consider the braid groups $\mathbf{B}_n(X)$ on finite simplicial complexes $X$, which are generalizations of those on both manifolds and graphs that have been studied already by many authors. We figure out the relationships between…

Let $G$ be the split orthogonal group of degree $2n$ over an arbitrary infinite field $\mathbb{F}$ of chararcteristic not $2$. In this paper, we classify multiple flag varieties $G/P_1\times\cdots\times G/P_k$ of finite type. Here a…

Let $G=Sp_{2r}({\mathbb C})$ be a simply connected simple algebraic group over $\mathbb{C}$ of type $C_r$, $B$ and $B_-$ be its two opposite Borel subgroups, and $W$ be the associated Weyl group. For $u$, $v\in W$, it is known that the…

We classify mutation-finite cluster algebras with arbitrary coefficients of geometric type.

We describe all Poisson brackets compatible with the natural cluster algebra structure in the open Schubert cell of the Grassmannian $G_k(n)$ and show that any such bracket endows $G_k(n)$ with a structure of a Poisson homogeneous space…

We study a subset of square free positive braids and we give a few algebraic characterizations of them and one geometric characterization: the set of positive braids whose closures are unlinks. We describe canonical forms of these braids…

We try to give a cluster algebraic interpretation of complex volume of knots. We construct the R-operator from the cluster mutations, and we show that it is regarded as a hyperbolic octahedron. The cluster variables are interpreted as edge…

We show that every trivial 3-strand braid diagram contains a disk, defined as a ribbon ending in opposed crossings. Under a convenient algebraic form, the result extends to every Artin--Tits group of dihedral type, but it fails to extend to…

A linear Gr-category is a category of finite-dimensional vector spaces graded by a finite group together with natural tensor product. We classify the braided monoidal structures of a class of linear Gr-categories via explicit computations…

An infinitary version of braid groups has been considered as a direct limit of n-braid groups. However, we can imagine more complicated braids with infinitely many strings. We invetisgate basic properties especially when the number of…

Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$, $B$ and $B_-$ be its two opposite Borel subgroups. For two elements $u$, $v$ of the Weyl group $W$, it is known that the coordinate ring ${\mathbb C}[G^{u,v}]$ of the…

The classification of Grassmannian cluster algebras resembles that of regular polygonal tilings. We conjecture that this resemblance may indicate a deeper connection between these seemingly unrelated structures.

We define pseudo-Garside groups and prove a theorem about them parallel to Garside's result on the word problem for the usual braid groups. The main novelty is that the set of simple elements can be infinite. We introduce a group B=B(Z^n)…

We study vector bundles on flag varieties over an algebraically closed field $k$. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\leq n-d)$ to be the Grassmannian manifold parameterizing linear subspaces of dimension $d$ in $k^n$, where…

This paper investigates a non simply-laced version of cluster structures for 2-Calabi-Yau or stably 2-Calabi-Yau categories over arbitrary fields. It results that 2-Calabi-Yau or stably 2-Calabi-Yau categories having a cluster tilting…

This paper is concerned with detecting when a closed braid and its axis are 'mutually braided' in the sense of Rudolph. It deals with closed braids which are fibred links, the simplest case being closed braids which present the unknot. The…