A basis for the diagonally signed-symmetric polynomials

Let n>0 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},…,x_{n},y_{1},…,y_{n}] by signed permutations simultaneously on both of the sets of variables x_{1},…,x_{n} and y_{1},…,y_{n}. The invariant ring M^{B_{n}}:=Q[x_{1},…,x_{n},y_{1},…,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},…, x^{2}{n} and y{1}^{2},…, y^{2}_{n} using signed descent monomials.