  We give a new type of Schur-Weyl duality for the representations of a family
of quantum subgroups and their centralizer algebra. We define and classify
singly-generated, Yang-Baxter relation planar algebras. We present the skein
theoretic construction of a new parameterized planar algebra. We construct
infinitely many new subfactors and unitary fusion categories, and compute their
trace formula as a closed-form expression, in terms of Young diagrams.
