Explicit determinantal formula for a class of banded matrices

In this short note, we provide a brief proof for a recent determinantal formula involving a particular family of banded matrices.

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 $\delta_{r}$, $r \geq n-1$.

Adjoint Mappings and Inverses of Matrices

In this paper, we investigate a general and determinantal representation, and conditions for the existence of a nonzero {2}-inverse X of a given complex matrix A. We introduce a determinantal formula for X, representing its elements in terms of minors of order s = rank (X), 1 ≤ s ≤ r = rank (A), taken from the matrix A and two adequately selected matrices. In accordance with these results, we find restrictions of the adjoint mapping such that the set A{2} is equal to the union of their images. Minors of {2}-inverses are also investigated. Restrictions to the set of {1, 2}-inverses produce the known results from [1–3, 10]. Also, in a partial case, we get known results from [11] relative to the Drazin inverse.