Robinson–Schensted Correspondence for the Walled Brauer Algebras and the Walled Signed Brauer Algebras

Abstract

In this paper, we develop a Robinson–Schensted algorithm for the walled Brauer algebras which gives the bijection between the walled Brauer diagram d and the pairs of standard tri-tableaux of shape \(\lambda =(\lambda _1,\lambda _2,\lambda _3)\) with \(\lambda _1=(2^{f}),\lambda _2 \vdash r-f\) and \(\lambda _3 \vdash s-f,\) for \(0 \le f \le \min (r,s).\) As a biproduct, we define a Robinson–Schensted correspondence for the walled signed Brauer algebras which gives the correspondence between the walled signed Brauer diagram d and the pairs of standard signed-tri-tableaux of shape \(\lambda =(\lambda _1,\lambda _2,\lambda _3)\) with \(\lambda _1=(2^{2f}),\lambda _2 \vdash _b r-f\) and \(\lambda _3 \vdash _b s-f,\) for \(0 \le f \le \min (r,s).\) We also derive the Knuth relations and the determinantal formula for the walled Brauer and the walled signed Brauer algebras by using the Robinson–Schensted correspondence.