Der Grad erzeugender Funktionen von Invariantenringen. (The degree of generating functions of rings of invariants)
type: Item
instance of: scholarly article
title: Der Grad erzeugender Funktionen von Invariantenringen. (The degree of generating functions of rings of invariants)
zbMATH Open document ID: 0635.20017
DOI: 10.1007/BF01163656
author: Friedrich Knop
author: Peter Littelmann
published in: Mathematische Zeitschrift
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173876
review text: Let G be a semisimple group defined over an algebraically closed field of characteristic zero and let V be a faithful representation of G. Denote by \(a_ d\) the dimension of the space of G-invariant polynomials on V which are homogeneous of degree d. It is known that the generating function \(\chi (t):=\sum^{\infty}_{d=0}a_ dt\) d is the power series expansion of a rational function f/g, f,g\(\in {\mathbb{Z}}[t]\), and \(\chi\) (t) fulfills the functional equation \(\chi (t^{-1})=\pm t\) \(q\chi\) (t) with \(q=\deg g-\deg f\). The first named author has shown [in Manuscr. Math. 56, 419-427 (1986; Zbl 0585.14033)] that \(q\leq \dim V\) and equality holds iff the set \(Z(V):=\{v\in V|\) dim \(G_ V>0)\}\) has codimension at least two. In this paper we classify all representations of simple groups and all irreducible representations of semisimple groups which do not have this property. For these representations we determine the corresponding q.
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14L24
zbMATH DE Number: 4033996
zbMATH Keywords: invariant polynomials
zbMATH Keywords: generating function
zbMATH Keywords: irreducible representations
zbMATH Keywords: semisimple groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4033996