Higher Specht polynomials for the complex reflection group \(G(r,p,n)\)
Suppose \(r,p,n\in\mathbb{N}\) (non-zero), with \(p\mid r\). Then the imprimitive complex reflection group \(G=G(r,p,n)\) [see \textit{G. C. Shephard} and \textit{J. A. Todd}, Can. J. Math. 6, 274-304 (1954; Zbl 0055.14305)] is the group of monomial matrices whose non-zero entries are of the form \(\zeta^j\) (\(0\leq j\leq r\)) and such that the \(d\)-th power of the product of all the non-zero entries is \(1\) (where \(d=r/p\)). Special cases include the Weyl groups of type \(A_n\), \(B_n\), \(D_n\) and \(G_2\). Denote by \(G(r,n)\) the same group of monomial matrices, without the restriction on the product of the non-zero entries. This group acts naturally on the polynomial ring \(P\) in \(n\) indeterminates, permuting the indeterminates and multiplying them by the roots of unity appearing in the matrix. The fundamental invariants under this action are given by the elementary symmetric functions in \(x_1^r,x_2^r,\ldots,x_n^r\). The coinvariant algebra, \(R=P/{J^+}\), where \(J^+\) is the ideal of \(P\) generated by the fundamental invariants, is isomorphic to \(\mathbb{C} G(r,n)\) as left module. The irreducible components of \(R\) have been described by \textit{S. Ariki, T. Terasoma} and \textit{H.-F. Yamada} [Hiroshima Math. J. 27, No. 1, 177-188 (1997; Zbl 0886.20009)] and \textit{T. Terasoma} and \textit{H.-F. Yamada} [Proc. Japan Acad., Ser. A, 69, No. 2, 41-44 (1993; Zbl 0811.20011)] (the latter for the case \(r=1\)) in terms of the higher Specht polynomials. The components are parametrized by \(r\)-tuples of Young diagrams with total number of squares equal to \(n\), and bases (the higher Specht polynomials) are given in terms of standard Young tableaux. In this paper, the authors consider the restriction of the above action to \(G(r,p,n)\). In this case, the corresponding coinvariant algebra is again isomorphic to the left regular representation. They generalize the higher Specht polynomials to \(G(r,p,n)\), proving that the resulting polynomials again provide bases for the irreducible representations. To do this, they apply Clifford theory, following \textit{J. R. Stembridge} [Pac. J. Math 140, No. 2, 353-396 (1989; Zbl 0641.20011)], to the normal subgroup \(G(r,p,n)\) of \(G(r,n)\) -- note that the quotient is cyclic.
scientific article; zbMATH DE number 1247668