A Basis for the Diagonally Signed-Symmetric Polynomials

Let n > 1 be an integer and let Bn denote the hyperoctahedral group of rank n. The group Bn acts on the polynomial ring Q[x1, . . . , xn, y1, . . . , yn] by signed permutations simultaneously on both of the sets of variables x1, . . . , xn and y1, . . . , yn. The invariant ring MBn := Q[x1, . . . , xn, y1, . . . , yn]n is the ring of diagonally signedsymmetric polynomials. In this article, we provide an explicit free basis of MBn as a module over the ring of symmetric polynomials on both of the sets of variables x1, . . . , x 2 n and y 2 1, . . . , y 2 n using signed descent monomials.