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 PnA with a canonical basis, BnA, 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.