q-Pascal's triangle and irreducible representations of the braid group B_3 in arbitrary dimension

TL;DR

This paper introduces a new family of irreducible representations of the braid group B_3 in any dimension, using a q-deformed Pascal triangle, extending previous classical and quantum approaches.

Contribution

It generalizes prior constructions by Humphries, Ferrand, Tuba, and Wenzl, providing a broader family of representations and connecting them to quantum group modules.

Findings

Constructed a ([(n+1)/2]+1)-parameter family of irreducible B_3 representations in arbitrary dimensions.

Extended classical Pascal triangle methods to a q-deformation framework.

Established links between the new representations and quantum group modules.

Abstract

We construct a [(n+1)/2]+1 parameters family of irreducible representations of the Braid group B_3 in arbitrary dimension n\in N, using a q-deformation of the Pascal triangle. This construction extends in particular results by S.P.Humphries [8], who constructed representations of the braid group B_3 in arbitrary dimension using the classical Pascal triangle. E.Ferrand [7] obtained an equivalent representation of B_3 by considering two special operators in the space C^n[X]. Slightly more general representations were given by I.Tuba and H.Wenzl [11]. They involve [(n+1)/2] parameters (and also use the classical Pascal triangle). The latter authors also gave the complete classification of all simple representations of B_3 for dimension n\leq 5. Our construction generalize all mentioned results and throws a new light on some of them. We also study the irreducibility and the equivalence of the representations. In [17] we establish the connection between the constructed representation of the braid group B_3 and the highest weight modules of U(sl_2) and quantum group U_q(sl_2).