Related papers: Graded cellular basis and Jucys-Murphy elements fo…
We study analogues of Jucys-Murphy elements in cellular algebras arising from repeated Jones basic constructions. Examples include Brauer and BMW algebras and their cyclotomic analogues.
We give a simple construction of Markov traces for Iwahori-Hecke algebras associated with infinite series of crystallographic Coxeter groups. In types B and D it is new, and generalizes a known construction in type A employing symmetric…
Two decades ago P. Martin and D. Woodcock made a surprising and prophetic link between statistical mechanics and representation theory. They observed that the decomposition numbers of the blob algebra (that appeared in the context of…
We define a method which produces explicit cellular bases for algebras obtained via a Jones basic construction. For the class of algebras in question, our method gives formulas for generic Murphy--type cellular bases indexed by paths on…
We first give a direct proof of a basis theorem for the cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}(q).$ Our approach follows Kleshchev's, which does not use the representation theory of $Y_{r,n}^{d}(q),$ and so it is very different from…
We settle a long-standing problem in the theory of Hecke algebras of complex reflection groups by constructing many (graded) integral cellular bases of these algebras. As applications, we explicitly construct the simple modules of Ariki's…
We consider a tower of generalized rook monoid algebras over the field $\mathbb{C}$ of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys-Murphy…
We show that the Temperley-Lieb algebra of type $A$ and the blob algebra (also known as the Temperley-Lieb algebra of type $ B$) at roots of unity are $ \mathbb Z$-graded algebras.We moreover show that they are graded cellular algebras,…
We give an explicit graded cellular basis of the $\mathfrak{sl}_3$-web algebra $K_S$. In order to do this, we identify Kuperberg's basis for the $\mathfrak{sl}_3$-web space $W_S$ with a version of Leclerc-Toffin's intermediate crystal basis…
We establish a framework for cellularity of algebras related to the Jones basic construction. Our framework allows a uniform proof of cellularity of Brauer algebras, ordinary and cyclotomic BMW algebras, walled Brauer algebras, partition…
We construct a faithful tensor representation for the Yokonuma-Hecke algebra Y, and use it to give a concrete isomorphism between Y and Shoji's modified Ariki-Koike algebra. We give a cellular basis for Y and show that the Jucys-Murphy…
This paper constructs an explicit homogeneous cellular basis for the cyclotomic Khovanov--Lauda--Rouquier algebras of type $A$.
In this paper we define an explicit basis for the $\mathfrak{gl}_n$-web algebra $H_n(\vec{k})$ (the $\mathfrak{gl}_n$ generalization of Khovanov's arc algebra) using categorified $q$-skew Howe duality. Our construction is a…
In this paper, we define a quotient of the cyclotomic Hecke algebra of type $G(r,1,n)$ as a generalisation of the Temperley-Lieb algebras of type $A$ and $B$. We establish a graded cellular structure for the generalised Temperley-Lieb…
We construct gradings on the simple modules of 2-boundary Temperley--Lieb algebras and symplectic blob algebras by realising the latter algebras as quotients of Varagnolo--Vasserot's orientifold quiver Hecke algebras. We prove that the…
In this paper, we construct semisimple deformations for cyclotomic quiver Hecke-Clifford superalgebras of types $A^{(1)}_{s-1}$, $C^{(1)}_{s}$, $A^{(2)}_{2s}$, $D^{(2)}_{s}$. We derive a unified dimension formula for the bi-weight spaces…
Let R be an integral domain and A a cellular algebra. Suppose that A is equipped with a family of Jucys-Murphy elements which satisfy the separation condition. Let K be the field of fractions of R. We give a necessary and sufficient…
The Jucys-Murphy elements for wreath products G_n associated to any finite group G are introduced and they play an important role in our study on the connections between class algebras of G_n for all n and vertex algebras. We construct an…
We study some quadratic algebras which are appeared in the low-dimensional topology and Schubert calculus. We introduce the Jucys-Murphy elements in the braid algebra and in the pure braid group, as well as the Dunkl elements in the…
This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer's cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a…