Related papers: Braid group and $q$-Racah polynomials
We introduce classical and non-deterministic finite automata associated with representations of the braid group. After briefly reviewing basic definitions on finite automata, Coxeter's groups and the associated word problem, we turn to the…
Ferroni and Larson gave a combinatorial interpretation of the braid Kazhdan-Lusztig polynomials in terms of series-parallel matroids. As a consequence, they confirmed an explicit formula for the leading Kazhdan-Lusztig coefficients of braid…
A presentation of the centralizer of the three-fold tensor product of the spin $s$ representation of the quantum group $U_q(\mathfrak{sl}_2)$ is provided. It is expressed as a quotient of the Askey-Wilson braid algebra. This newly defined…
We propose a definition by generators and relations of the rank $n-2$ Askey-Wilson algebra $\mathfrak{aw}(n)$ for any integer $n$, generalising the known presentation for the usual case $n=3$. The generators are indexed by connected subsets…
Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution -- that of the antiparallel…
The Gasper and Rahman multivariate $(-q)$-Racah polynomials appear as connection coefficients between bases diagonalizing different abelian subalgebras of the recently defined higher rank $q$-Bannai-Ito algebra $\mathcal{A}_n^q$. Lifting…
We describe the structure of the irreducible representations of crossed products of unital C*-algebras by actions of finite groups in terms of irreducible representations of the C*-algebras on which the groups act. We then apply this…
We study a wide range of homologically-defined representations of surface braid groups and of mapping class groups of surfaces, including the Lawrence-Bigelow representations of the classical braid groups. These representations naturally…
This paper is the first part of a series of papers aimed at improving the classification by Formanek of the irreducible representations of Artin braid groups of small dimension. In this paper we classify all the irreducible complex…
In this paper the author finds explicitly all finite-dimensional irreducible representations of a series of finite permutation groups that are homomorphic images of Artin braid group.
We consider subgroups of the braid groups which are generated by $k$-th powers of the standard generators and prove that any infinite intersection (with even $k$) is trivial. This is motivated by some conjectures of Squier concerning the…
An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown…
By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a…
Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang-Baxter equation, which is precisely the Yang-Baxter equation satisfied by 6j-symbols. We investigate one of…
We study solutions of the Knizhnik-Zamolodchikov equation for discrete representations of SU(2)_k at rational level k+2=p/q using a regular basis in which the braid matrices are well defined for all spins. We show that at spin J=(j+1)p-1…
We use the relation between the quantum su(2) R-matrix and the Burau representation of the braid group in order to study the structure of the colored Jones polynomial of links. We show that similarly to the case of a knot, the colored Jones…
We study and give examples of braided groupoids, and, a fortiori, non-degenerate solutions of the quiver-theoretical braid equation.
We establish a link between the new theory of $q$-deformed rational numbers and the classical Burau representation of the braid group $\mathrm{B}_3$. We apply this link to the open problem of classification of faithful complex…
Bivariate Griffiths polynomials of Racah type are constructed from univariate Racah polynomials. The bispectral properties of the former are deduced from simple properties of the latter. A duality relation and the orthogonality of these…
We investigate braid group representations associated with unitary braided vector spaces, focusing on a conjecture that such representations should have virtually abelian images in general and finite image provided the braiding has finite…