A Basis for the Diagonally Signed-Symmetric Polynomials

Let $n\ge 1$ be an integer and let $B_{n}$ denote the hyperoctahedral group of rank $n$. The group $B_{n}$ acts on the polynomial ring $Q[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]$ by signed permutations simultaneously on both of the sets of variables $x_{1},\dots,x_{n}$ and $y_{1},\dots,y_{n}.$ The invariant ring $M^{B_{n}}:=Q[x_{1},\dots,x_{n},y_{1},\dots,y_{n}]^{B_{n}}$ is the ring of diagonally signed-symmetric polynomials. In this article, we provide an explicit free basis of $M^{B_{n}}$ as a module over the ring of symmetric polynomials on both of the sets of variables $x_{1}^{2},\dots, x^{2}_{n}$ and $y_{1}^{2},\dots, y^{2}_{n}$ using signed descent monomials.