Gelfand W-graphs for classical Weyl groups

A Gelfand model for an algebra is a module given by direct sum of irreducible submodules, with every isomorphism class modules represented exactly once. We introduce the notion perfect finite Coxeter group, which certain set discrete data (involving Rains and Vazirani's concept involution) that parametrizes associated Iwahori-Hecke algebra. describe models all classical Weyl groups, excluding type D in even rank. The representations attached to these simultaneously generalize constructions Adin, Postnikov, Roichman (from other types) Araujo Bratten group algebras algebras). show each derived from has canonical basis gives rise pair related $W$-graphs, we call $W$-graphs. For types BC D, prove $W$-graphs are dual other, phenomenon does not occur A.