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

• Published in: Journal of algebra 1995-04, Vol.173 (1), p.122-143
• Main Author: Allen, E.E.
• Format: Article
• Language: English
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1006/jabr.1995.1080

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 PSC 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 PSC has the remarkable property that when restricted to R*D, exactly one-half of the elements of PSC are non-zero and the non-zero polynomials PSCD 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 PSCD gives the decomposition of R*D into its irreducible components when n is odd. A slight modification of PSCD 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.