A -BASIS OF THE -INVARIANT SUBRING OF

Let be the subring of the polynomial ring of independent variables generated by the differences . Let be the invariant subring of under the action of the symmetric group of degree defined by . We investigate and construct a -basis of the submodules of of degree under certain conditions. Furthermore, is expressed as the kernel of the differential operator of the -invariant subring for elementary symmetric polynomials of degree of . With respect to an ordered -basis of , we determine the exponents on the minimal terms of the -basis vectors of (Theorem A).