On planar algebras arising from hypergroups

Let $A$ be an associative algebra with identity and with trace. We study the family of planar algebras on 1-boxes that arise from $A$ in the work of Jones, but with the added assumption that the labels on the 1-boxes come from a discrete hypergroup in the sense of Sunder. This construction equips the algebra $P_n^A$ with a canonical basis, ${\text{BB}}_n^A$, which turns out to be a ``tabular’’ basis. We examine special cases of this construction to exhibit a close connection between such bases and Kazhdan–Lusztig bases of Hecke algebras of types $A$, $B$, $H$ or $I$.