Robinson-Schensted Correspondence for the Signed Brauer Algebras

In this paper, we develop the Robinson-Schensted correspondence for the signed Brauer algebra. The Robinson-Schensted correspondence gives the bijection between the set of signed Brauer diagrams d and the pairs of standard bi-domino tableaux of shape λ = (λ1, λ2) with λ1 = (2 2f ), λ2 ∈ Γf,r where Γf,r = {λ|λ ` 2(n − 2f) + |δr| whose 2−core is δr, δr = (r, r − 1, . . . , 1, 0)}, for fixed r ≥ 0 and 0 ≤ f ≤ [ n 2 ] . We also give the Robinson-Schensted for the signed Brauer algebra using the vacillating tableau which gives the bijection between the set of signed Brauer diagrams V n and the pairs of d-vacillating tableaux of shape λ ∈ Γf,r and 0 ≤ f ≤ [ n 2 ] . We derive the Knuth relations and the determinantal formula for the signed Brauer algebra by using the Robinson-Schensted correspondence for the standard bi-dominotableau whose core is δr, r ≥ n− 1.