Related papers: Fusion and specialization for type ADE shuffle alg…
Let Q be a quiver of type ADE. We construct the corresponding Auslander-Reiten quiver as a topological complex inside the Coxeter complex associated with the underlying Dynkin diagram. We use the notion of chamber weights coming from the…
The commutative trigonometric shuffle algebra ${\mathrm A}$ is a space of symmetric rational functions satisfying certain wheel conditions. We describe a ring isomorphism between ${\mathrm A}$ and the center of the Hecke algebra using a…
The initial classification of fusion rules have shown that rational conformal field theory is very limited. In this paper we study the fusion rules of extend ed current algebras. Explicit formulas are given for the S matrix and the fusion…
Let Q be a Dynkin quiver of type A. The bounded derived category of the path algebra of Q has an autoequivalence given by the composition of the Auslander-Reiten translate and the square of the shift functor. We classify the maximal rigid…
We present a simple unified formula expressing the denominators of the normalized R-matrices between the fundamental modules over the quantum loop algebras of type ADE. It has an interpretation in terms of representations of the Dynkin…
We provide an explicit Dynkin diagrammatic description of the c-vectors and the d-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix. We use the surface…
We first provide an explicit combinatorial description of the Auslander-Reiten quiver $\Gamma^Q$ of finite type $D$. Then we can investigate the categories of finite dimensional representations over the quantum affine algebra…
In this short note, we show that the Ginzburg-Vasserot map between the quantum affine algebra of type A_(n-1) and the equivariant K-theory group of the Steinberg Variety (of n-step flags in C^d) restricts and remains surjective at the level…
We classify spherical modules and full exceptional sequences of modules over the Auslander algebra of $k[x]/(x^t)$. We categorify the left and right symmetric group actions on these exceptional sequences to two braid group actions: of…
We construct and study new generalisations to rooted trees and forests of some properties of shuffles of words. First, we build a coproduct on rooted trees which, together with their shuffle, endow them with bialgebra structure. We then…
Let $Q$ be a Dynkin quiver and $\Pi$ the corresponding set of positive roots. For the preprojective algebra $\Lambda$ associated to $Q$ we produce a rigid $\Lambda$-module $I_Q$ with $r=|\Pi|$ pairwise non-isomorphic indecomposable direct…
In this note we describe a general elementary procedure to attach a fusion ring to any Kac-Moody algebra of affine type. In the case of untwisted affine algebras, they are usual fusion rings in the literature. In the case of twisted affine…
We obtain a presentation of the t-deformed Grothendieck ring of a quantum loop algebra of Dynkin type A, D, E. Specializing t at the the square root of the cardinality of a finite field F, we obtain an isomorphism with the derived Hall…
These graphs generalize both affine and finite type and provide an explanation for some quantum properties of roots of unity.
We study double affine Hecke algebras at roots of unity and their relations with deformed Hilbert schemes.
In this article, we construct certain commutative subalgebras of the big shuffle algebra of cyclic type. This can be considered as a generalization of the similar construction for the small shuffle algebra, obtained by…
We consider a class of quasi-Hopf algebras which we call \emph{generalized twisted quantum doubles}. They are abelian extensions $H = \mb{C}[\bar{G}] \bowtie \mb{C}[G]$ ($G$ is a finite group and $\bar{G}$ a homomorphic image), possibly…
As a continuation of the previous paper, we find a combinatorial interpretation of Dorey's rule for type $C_n$ via twisted Auslander-Reiten quivers (AR-quivers) of type $D_{n+1}$, which are combinatorial AR-quivers related to certain Dynkin…
We introduce an operation on skew-symmetric matrices over $\mathbb{Z}/\ell\mathbb{Z}$ called switching, and also define a class of skew-symmetric matrices over $\mathbb{Z}/\ell\mathbb{Z}$ referred to as modular Eulerian matrices. We then…
We show that the Ocneanu algebra of quantum symmetries, for an ADE diagram (or for higher Coxeter-Dynkin systems, like the Di Francesco - Zuber system) is, in most cases, deduced from the structure of the modular T matrix in the A series.…