New bases for the decomposition of the graded left regular representation of the reflection groups of type \(B_ n\) and \(D_ n\)
Let \(\mathbb{R}=Q[x_1, x_2, \dots, x_n]\) be the ring of polynomials in the variables \(x_1, x_2, \dots, x_n\). 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 \(\mathbb{R}^*_B\) denote the quotient of \(\mathbb{R}\) by the ideal generated by \(I_B\). It is well known [see \textit{I. G. Macdonald}, Finite reflection groups, UCSD classroom notes, La Jolla, CA (1991)] that the action of \(W_B\) on the quotient ring \(\mathbb{R}^*_B\) is isomorphic to the left regular representation of \(W_B\). Using methods similar to those of \textit{E. E. Allen} [Proc. Natl. Acad. Sci. USA 89, No. 9, 3980-3984 (1992; Zbl 0767.05095) and Adv. Math. 100, 262-292 (1993; Zbl 0795.20006)] the author constructs a basis \({\mathcal P} {\mathcal S} {\mathcal C}\) of \(\mathbb{R}^*_B\) which exhibits the decomposition of \(\mathbb{R}^*_B\) into its irreducible components. Now let \(W_D\) be the finite reflection group of type \(D_n\), let \(I_D\) be a basic set of invariants for \(W_D\), and let \(\mathbb{R}^*_D\) be the quotient of \(\mathbb{R}\) with the ideal generated by \(I_D\). The author shows that the basis \({\mathcal P} {\mathcal S} {\mathcal C}\) has the remarkable property that when restricted to \(\mathbb{R}^*_D\), exactly one-half of the elements of \({\mathcal P} {\mathcal S} {\mathcal C}\) are non-zero and the non-zero polynomials \({\mathcal P} {\mathcal S} {\mathcal C}_D\) form a basis for \(\mathbb{R}^*_D\). The action of \(W_D\) on the quotient ring \(\mathbb{R}^*_D\) is also isomorphic to the left regular representation of \(W_D\). This collection of polynomials \({\mathcal P} {\mathcal S} {\mathcal C}_D\) gives the decomposition of \(\mathbb{R}^*_D\) into its irreducible components when \(n\) is odd. A slight modification of \({\mathcal P} {\mathcal S} {\mathcal C}_D\) gives a basis for the decomposition of \(\mathbb{R}^*_D\) when \(n\) is even. The author uses these bases to construct the respective graded characters of \(\mathbb{R}^*_B\) and \(\mathbb{R}^*_D\).
scientific article; zbMATH DE number 763937