Related papers: Floer potentials, cluster algebras and quiver repr…
We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system.
We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on…
This is an introduction to cluster algebras and their common triangular bases. These bases are Kazhdan-Lusztig-type and serve as the canonical bases of cluster algebras from the representation-theoretic point of view. We review seeds…
We give an axiomatic framework for studying the representation theory of towers of algebras. We introduce a new class of algebras, contour algebras, generalising (and interpolating between) blob algebras and cyclotomic Temperley-Lieb…
We give an introduction to the theory of cluster categories and cluster tilted algebras. We include some background on the theory of cluster algebras, and discuss the interplay with cluster categories and cluster tilted algebras.
We compute the quiver of any monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term…
We explain how to use bordered algebras to compute a version of link Floer homology. As a corollary, we also give a fast computation of the Thuston polytope for links in the three-sphere.
We study rank-2 cluster scattering diagrams through moduli spaces of quiver representations and a recently developed combinatorial framework of tight gradings. Combining quiver-theoretic and combinatorial methods, we prove and extend a…
In this note we explain how to obtain cluster algebras from triangulations of (punctured) discs following the approach of S. Fomin, M. Shapiro and D. Thurston. Furthermore, we give a description of m-cluster categories via diagonals (arcs)…
We describe a construction of fuzzy spaces which approximate projective toric varieties. The construction uses the canonical embedding of such varieties into a complex projective space: The algebra of fuzzy functions on a toric variety is…
Toric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. An upper bound on the volume and on the number of boundary lattice points of these polygons is…
We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. By using $d-$cluster categories which are defined by Keller as triangulated orbit categories of (bounded) derived…
We give a survey of how the relatively young theory of operator spaces has led to a deeper understanding of the Fourier algebra of a locally compact group (and of related algebras).
In this lecture, we survey a number of recent results and developments regarding the representation theory of infinite-dimensional quantum groups (quantum affine algebras and related algebras), as well as their connections with cluster…
For any rigid presentation $e$, we construct an orthogonal projection functor to ${\rm rep}(e^\perp)$ left adjoint to the natural embedding. We establish a bijection between presentations in ${\rm rep}(e^\perp)$ and presentations compatible…
We study relations between cluster algebra invariants and link invariants. First, we show that several constructions of positroid links (permutation links, Richardson links, grid diagram links, plabic graph links) give rise to isotopic…
We prove a multiplication theorem for quantum cluster algebras of acyclic quivers. The theorem generalizes the multiplication formula for quantum cluster variables in \cite{fanqin}. We apply the formula to construct some $\mathbb{ZP}$-bases…
An umbral type formalism is used to derive integrals involving products of Laguerre polynomials and other special functions.
We prove that Floer cohomology of cyclic Lagrangian correspondences is invariant under transverse and embedded composition of Lagrangians under a general set of assumptions. In the Corrigendum, we introduce an additional assumption of…
We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality…