Generalized hypergeometric series for Racah matrices in rectangular representations

TL;DR

This paper explores the extension of hypergeometric series representations of Racah matrices from symmetric to more general representations of quantum groups, involving complex sums over Young diagrams and new mathematical structures.

Contribution

It introduces a generalized hypergeometric series framework for Racah matrices in arbitrary $SU_q(N)$ representations, expanding beyond symmetric cases.

Findings

Racah matrices for symmetric representations relate to Askey-Wilson polynomials.

Generalized series involve multiple sums over Young diagrams with non-factorized quantities.

New structures include skew characters and Littlewood-Richardson weights.

Abstract

One of spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group $SU_q(2)$ through the Askey-Wilson polynomials, associated with the $q$-hypergeometric functions ${_4\phi_3}$. Recently it was shown that this is in fact the general property of symmetric representations, valid for arbitrary $SU_q(N)$, 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 $SU_q(N)$. 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.