Generalized exponents and forms.
type: Item
instance of: scholarly article
title: Generalized exponents and forms.
zbMATH Open document ID: 1086.20021
author: Anne V. Shepler
published in: Journal of Algebraic Combinatorics
publication date: 2005-06-08T00:00:00Z
review text: Let \(V=\mathbb{C}^l\) or \(\mathbb{R}^l\) be a complex or real vector space. A reflection is an element of \(\text{GL}(V)\) whose fixed point set is a hyperplane in \(V\). Let \(G\) be a reflection group, i.e., a finite subgroup of \(\text{GL}(V)\) generated by reflections. We assume all \(G\)-modules are \(\mathbb{C} G\)-modules. For any \(G\)-module \(U\) and irreducible \(G\)-module \(M\), let \(U^M\) be the isotypic component of \(U\) of type \(M\), i.e., the direct sum of those \(G\)-submodules of \(U\) isomorphic to \(M\). Let \(U^G:=\{u\in U:gu=u\) for all \(g\in G\}\) denote the set of \(G\)-invariants. The reflection group \(G\) acts contragradiently on \(V^*\) and thus on the symmetric algebra \(S:=S(V^*)\), which we identify with the algebra of polynomial functions on \(V\). Let \(I\subset S\) be the Hilbert ideal generated by the invariant polynomials of positive degree. Then \(S^G=\mathbb{C}[f_1,\dots,f_l]\) for some homogeneous polynomials \(f_1,\dots,f_l\) called basic invariants. The algebra \(S/I\) is called the coinvariant algebra. \(S/I\) is isomorphic to the regular representation and \(S\simeq S^G\otimes S/I\) as \(G\)-modules. For any irreducible \(G\)-module \(M\), the isotypic component \((S/I)^M\) decomposes as \(M_1\oplus M_2\oplus\cdots\oplus M_{\dim M}\) for some homogeneous subspaces \(M_i\simeq M\) of degree \(e_i(M)\). We call \(e_1(M),e_2(M),\dots,e_{\dim M}(M)\) the \(M\)-exponents. Let \(m_1,\dots,m_l\) be the \(V\)-exponents, called the exponents of the group. Similarly, let \(m_1^*,\dots,m_l^*\) be the \(V^*\)-exponents, called the coexponents of the group. The exponents and coexponents of the group indicate the invariant theory of differential forms and derivations. In this paper, the author investigates twisted reflection representations (\(V\) tensor a linear character) using the theory of semi-invariant differential forms. The author uses information about semi-invariant polynomials, forms, and derivations to describe general…
reviewed by: Chen Chengdong
Mathematics Subject Classification ID: 20F55
Mathematics Subject Classification ID: 51F15
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 52C35
Mathematics Subject Classification ID: 32S22
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 13A50
zbMATH DE Number: 2174455
zbMATH Keywords: reflection groups
zbMATH Keywords: invariant theory
zbMATH Keywords: generalized exponents
zbMATH Keywords: Coxeter groups
zbMATH Keywords: fake degrees
zbMATH Keywords: hyperplane arrangements
zbMATH Keywords: derivations
MaRDI profile type: Publication
full work available at URL: s10801-005-6908-y
OpenAlex ID: W2046303284
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DOI: 10.1007/S10801-005-6908-Y
description: scientific article; zbMATH DE number 2174455