Generalized hypergeometric series for Racah matrices in rectangular representations

Abstract

One of spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group $S{U}_q(2)$ through the Askey-Wilson polynomials, associated with the $q$-hypergeometric functions ${}_4{\varphi }_3$. Recently it was shown that this is in fact the general property of symmetric representations, valid for arbitrary $S{U}_q(N)$, at least for exclusive Racah matrices $\bar{S}$. The natural question then is what substitutes the conventional $q$-hypergeometric polynomials when representations are more general? New advances in the theory of matrices $\bar{S}$, provided by the study of differential expansions of knot polynomials, suggest that these are multiple sums over Young sub-diagrams of the one, which describes the original representation of $S{U}_q(N)$. A less trivial fact is that the entries of the sum are not just the factorized combinations of quantum dimensions, as in the ordinary hypergeometric series, but involve non-factorized quantities, like the skew characters and their further generalizations -- as well as associated additional summations with the Littlewood-Richardson weights.

8 pages

References in corpus (7)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Racah matrices and hidden integrability in evolution of knots
• Differential expansion and rectangular HOMFLY for the figure eight knot
• Factorization of differential expansion for antiparallel double-braid knots
• On rectangular HOMFLY for twist knots
• Rectangular superpolynomials for the figure-eight knot
• On moduli space of symmetric orthogonal matrices and exclusive Racah matrix $\bar{S}$ for representation $R=[3,1]$ with multiplicities

Cited by in corpus (4)

• A slow review of the AGT correspondence
• Open String Amplitudes: Singularities, Asymptotics, and New Representations
• Cyclotomic expansions of HOMFLY-PT colored by rectangular Young diagrams
• Hypergeometry from $\hat{\mathrm{P}}$-Symmetry: Feynman Integrals in One and Two Dimensions