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

Wahl, Jonas¹

¹Hausdorff Center for Mathematics Endenicher Allee 62 D-53115 Bonn, Germany

Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 413-436

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 $ℂ\left[S_{\infty }\right]$ 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 $ℂ[S_{\infty }\times S_{\infty }]$. Our approach relies on methods developed by Vershik and Nikitin in [37].

Article information

Received: 2020-10-07
Revised: 2021-11-05
Accepted: 2021-12-27
Published online: 2022-07-07

DOI: 10.5802/alco.218

Classification: 20G42, 05E10, 05A18
Keywords: Branching graph, minimal boundary, trace simplex, easy quantum groups.

License: CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights