Robinson-Schensted correspondence for the signed Brauer algebras (Q1010603)

Summary: 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-dominotableaux of shape \(\lambda=(\lambda_1,\lambda_2)\) with \(\lambda_1=(2^{2f}),\lambda_2 \in \overline{\Gamma}_{f,r}\) where \(\overline{\Gamma}_{f,r}=\{ \lambda | \lambda\vdash 2(n-2f)+|\delta_r| \text{whose} \;2-\text{core}\, \text{is}\, \delta_{\text r}, \delta_{\text r}=(r,r-1,\dots,1,0)\}\), for fixed \(r\geq 0\) and \(0\leq f \leq [\frac{n}{2}]\). 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 \({\overline{V}_n}\) and the pairs of \(d\)-vacillating tableaux of shape \(\lambda \in \overline{\Gamma}_{f,r}\) and \(0\leq f \leq [\frac{n}{2}]\). 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 \(\delta_{r}, r \geq n-1\).