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-dominotableaux of shape $\lambda=(\lambda_1,\lambda_2)$ with $\lambda_1=(2^{2f}),\lambda_2 \in \overline{\Gamma}_{f,r}$ where $\overline{\Gamma}_{f,r}=\{ \lambda | \lambda\vdash 2(n-2f)+|\delta_r| {\rm \ whose } \ 2{\rm-core \ is \ \delta_r, \ } \delta_r=(r,r-1,\ldots,1,0)\}$, for fixed $r\geq 0$ and $0\leq f \leq \left[{n\over 2}\right]$. 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 $0\leq f \leq \left[{n\over 2}\right]$. 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 $\delta_{r}$, $r \geq n-1$.