New Bases for the Decomposition of the Graded Left Regular Representation of the Reflection Groups of Type Bn and Dn Original Research Article

Let R = Q[x1, x2, . . . , xn] be the ring of polynomials in the variables x1, x2, . . . , xn. Let WB be the finite reflection group of type Bn, let IB be a basic set of invariants of WB, and let R*B, denote the quotient of R by the ideal generated by IB. It is well known (see [Macdonald, 1991]) that the action of WB on the quotient ring R*B, viewed as a vector space over R, is isomorphic to the left regular representation of WB. Using methods similar to those in [Allen, 1992, 1993] we construct a basis imageimageimage of R*B which exhibits the decomposition of R*B into its irreducible components. Now let WD be the finite reflection group of type Dn, let ID be a basic set of invariants for WD, and let R*D be the quotient of R with the ideal generated by ID. We will show that the basis imageimageimage has the remarkable property that when restricted to R*D, exactly one-half of the elements of imageimageimage are non-zero and the non-zero polynomials imageimageimageD form a basis for R*D. The action of WD on the quotient ring R*D, viewed as a vector space of R, is also isomorphic to the left regular representation of WD (see [Macdonald, 1991]). This collection of polynomials imageimageimageD gives the decomposition of R*D into its irreducible components when n is odd. A slight modification of imageimageimageD gives a basis for the decomposition of R*D when n is even. We use these bases to construct the respective graded characters of R*B and R*D.