NEW BASES FOR THE DECOMPOSITION OF THE GRADED LEFT REGULAR REPRESENTATION OF THE REFLECTION GROUPS OF TYPE B-N AND D-N

Let R = Q[x(1),x(2),...,x(n)] be the ring of polynomials in the variab les x(1),x(2),...,x(n). Let W-B be the finite reflection group of type B-n, let I-B be a basic set of invariants of W-B, and let R(B), deno te the quotient of R by the ideal generated by I-B. It is well known ( see [Macdonald, 1991]) that the action of W-B on the quotient ring R(B ), viewed as a vector space over R, is isomorphic to the left regular representation of W-B. Using methods similar to those in [Allen, 1992 , 1993] we construct a basis PLC of R(B) which exhibits the decomposi tion of R(B) into its irreducible components. Now let W-D be the fini te reflection group of type D-n, let I-D be a basic set of invariants for W-D, and let R(D) be the quotient of R with the ideal generated b y I-D. We will show that the basis PLC has the remarkable property tha t when restricted to R(D), exactly one-half of the elements of PLC ar e non-zero and the non-zero polynomials PLC(D) form a basis for R(D). The action of W-D on the quotient ring R(D), viewed as a vector spac e of R, is also isomorphic to the left regular representation of W-D ( see [Macdonald, 1991]). This collection of polynomials PLC(D) gives th e decomposition of R(D) into its irreducible components when n is odd . A slight modification of PLC(D) gives a basis for the decomposition of R(D) when n is even. We use these bases to construct the respectiv e graded characters of R(B) and R(D)*. (C) 1995 Academic Press, Inc,