Higher Specht polynomials for the complex reflection group \(G(r,p,n)\)
type: Item
instance of: scholarly article
title: Higher Specht polynomials for the complex reflection group \(G(r,p,n)\)
zbMATH Open document ID: 0919.20027
author: K. Appert
published in: Hokkaido Mathematical Journal
publication date: 1999-07-06T00:00:00Z
review text: 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 pro…
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 20G20
zbMATH DE Number: 1247668
zbMATH Keywords: complex reflection groups
zbMATH Keywords: coinvariant rings
zbMATH Keywords: Clifford theory
zbMATH Keywords: tableaux
zbMATH Keywords: Weyl groups
zbMATH Keywords: symmetric functions
zbMATH Keywords: Young diagrams
zbMATH Keywords: irreducible representations
reviewed by: Bethany R. Marsh
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1351001459
OpenAlex ID: W2086585685
DOI: 10.14492/HOKMJ/1351001459
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description: scientific article; zbMATH DE number 1247668