Factorization of differential expansion for non-rectangular representations

Abstract

Factorization of the differential expansion (DE) coefficients for colored HOMFLY-PT polynomials of antiparallel double braids, originally discovered for rectangular representations [Formula: see text], in the case of rectangular representations [Formula: see text], is extended to the first non-rectangular representations [Formula: see text] and [Formula: see text]. This increases chances that such factorization will take place for generic [Formula: see text], thus fixing the shape of the DE. We illustrate the power of the method by conjecturing the DE-induced expression for double-braid polynomials for all [Formula: see text]. In variance with the rectangular case, the knowledge for double braids is not fully sufficient to deduce the exclusive Racah matrix [Formula: see text] — the entries in the sectors with nontrivial multiplicities sum up and remain unseparated. Still, a considerable piece of the matrix is extracted directly and its other elements can be found by solving the unitarity constraints.