Traces on diagram algebras II: centralizer algebras of easy groups and new variations of the Young graph

Abstract

In continuation of our recent work [39], we classify the extremal traces on infinite diagram algebras that appear in the context of Schur-Weyl duality for Banica and Speicher's easy groups [3]. We show that the branching graphs of these algebras describe walks on new variations of the Young graph which in turn explicate curious ways of growing Young diagrams. As a consequence, we prove that the extremal traces on generic rook-Brauer algebras are always extensions of extremal traces on the group algebra C[S∞] of the infinite symmetric group. Moreover, we conjecture that the same is true for generic parameter deformations of the centralizers of the hyperoctahedral group and we reduce this conjecture to a conceptually much simpler numerical statement. Lastly, we address the trace classification problem for the Schur-Weyl dual of the halfliberated orthogonal group O * N , in which case extremal traces are always extensions of extremal traces on C[S∞ ×S∞]. Our approach relies on methods developed by Vershik and Nikitin in [37].