Indian Journal of Science and Technology

Abstract

Objectives: This study aims to establish the Robinson-Schensted correspondence for the walled Klein-4 Brauer algebra. We define a bijection between walled Klein-4 Brauer diagrams and pairs of Klein-4 vacillating tableaux, providing a combinatorial proof for a significant identity related to this algebra. Method: We construct the Robinson-Schensted correspondence using a function, d → (P(λ,µ), Q(λ,µ)), mapping diagrams to tableaux by employing the Robinson-Schensted correspondence for the generalised Klein-4 group. This involves a pair of 4-Young standard tableaux and the paths of the Bratteli diagram. Findings: Our study yields a combinatorial proof of the identity (r+s) = , where represents the number of paths from the 0th stage of the Bratteli diagram to the shape (λ,µ), ={(λ,µ): λ is a 4-partition of (r−w), µ is a 4-partition of (s−w), 0 ≤ w ≤ min(r, s)} and r, s are non-negative integers. Novelty: By employing the Robinson-Schensted correspondence, we establish fundamental relations and rules, enriching the understanding of walled Klein-4 Brauer algebra in the realm of algebraic combinatorics.

Keywords: Brauer algebra, Partition algebra, 4-Partition, Walled Klein-4 Brauer algebra, Robinson-Schensted correspondence