Braid group and $q$-Racah polynomials

TL;DR

This paper explores the relationship between braid group representations and $q$-Racah polynomials in the context of $U_q( ext{sl}_2)$, providing explicit formulas and connecting algebraic structures with special functions.

Contribution

It establishes a conjugation relation between intermediate Casimir elements and braid group representations, expressing braid matrices explicitly via $q$-Racah polynomials.

Findings

Explicit formulas for braid group matrices in terms of $q$-Racah polynomials.

Connection between intermediate Casimir elements and braid group conjugation.

Derivation of $q$-Racah polynomial formulas from algebraic relations.

Abstract

The irreducible representations of two intermediate Casimir elements associated to the recoupling of three identical irreducible representations of $U_q(\mathfrak{sl}_2)$ are considered. It is shown that these intermediate Casimirs are related by a conjugation involving braid group representations. Consequently, the entries of the braid group matrices are explicitly given in terms of the $q$-Racah polynomials which appear as $6j$-symbols in the Racah problem for $U_q(\mathfrak{sl}_2)$. Formulas for these polynomials are derived from the algebraic relations satisfied by the braid group representations.