Related papers: Caldero-Chapoton algebras
We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals which are invariant under the action of the corresponding rational Cherednik algebra…
We show that skew-orthogonal functions, defined with respect to Jacobi weight $w_{a,b}(x)={(1-x)}^a{(1+x)}^b$, $a$, $b>-1$, including the limiting cases of Laguerre ($w_{a}(x)=x^{a}e^{-x}$, $a > -1$) and Gaussian weight ($w(x)=e^{-x^2}$),…
We study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by…
Clifford algebras are important structures in Geometric Algebra and Quantum Mechanics. They have allowed a formalization of the primitive operators in Quantum Theory. The algebras are built over vector spaces with dimension a power of 2…
In this paper we give a direct proof of the positivity conjecture for adapted quantum cluster variables. Moreover, our process allows one to explicitly compute formulas for all adapted cluster monomials and certain ordered products of…
Let $A = \Bbbk Q / I$ be the path algebra of any finite quiver $Q$ modulo any two-sided ideal $I$ of relations and let $R$ be any reduction system satisfying the diamond condition for $I$. We introduce an intrinsic notion of deformation of…
A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, i.e. subvarieties of the varieties of representations. The study of this monoid leads to interesting…
In accordance with P. Vogel, a set of algebra structures in Chern-Simons theory can be made universal, independent of a particular family of simple Lie algebras. In particular, this means that various quantities in the adjoint…
We introduce the \emph{universal algebra} of two Poisson algebras $P$ and $Q$ as a commutative algebra $A:={\mathcal P} (P, \, Q )$ satisfying a certain universal property. The universal algebra is shown to exist for any finite dimensional…
We study a new class of quiver algebras on surfaces, called 'geodesic ghor algebras'. These algebras generalize cancellative dimer algebras on a torus to higher genus surfaces, where the relations come from perfect matchings rather than a…
We introduce a notion of representation for a class of generalised quivers known as Coxeter quivers. These representations are built using fusion categories associated to $U_q(\mathfrak{s}\mathfrak{l}_2)$ at roots of unity and we show that…
Generalized Cluster Algebras (GCA) are generalizations of Cluster Algebras (CA) with higher-order exchange relations. Previously, Chekhov-Shapiro conjectured that every GCA can be embedded into a CA. In this paper, we prove a modified…
A $r$-parameter ${u}_{\{\kappa_1, \kappa_2, \cdots, \kappa_r\}}(2)$ algebra is introduced. Finite unitary representations are investigated. This polynomial algebra reduces via a contraction procedure to the generalized Weyl-Heisenberg…
This is a concise introduction to Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition of cluster algebras (geometric, without coefficients), construct…
The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a…
We investigate the representation theory of the rational and trigonometric Cherednik algebra of type $GL_n$ by means of combinatorics on periodic (or cylindrical) skew diagrams. We introduce and study standard tableaux and plane partitions…
We consider Dunkl version of Laplace-Runge-Lenz vector associated with a finite Coxeter group $W$ acting geometrically in $\mathbb R^N$ with multiplicity function $g$. This vector generalizes the usual Laplace-Runge-Lenz vector and its…
We attempt to explain the ubiquity of tableaux and of Pieri and Cauchy formulae for combinatorially defined families of symmetric functions. We show that such formulae are to be expected from symmetric functions arising from representations…
The Chebyshev potential of a K\"ahler potential on a projective variety, introduced by Witt Nystr\"om, is a convex function defined on the Okounkov body. It is a generalization of the symplectic potential of a torus-invariant K\"ahler…
The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field FF_q and any sequence ii of simple objects in C the element X_{V,ii}…