Elements of compact connected simple Lie groups with prime power order and given field of characters
type: Item
instance of: scholarly article
title: Elements of compact connected simple Lie groups with prime power order and given field of characters
zbMATH Open document ID: 0807.22005
DOI: 10.1007/BF01264522
published in: Geometriae Dedicata
publication date: 1993-05-16T00:00:00Z
review text: Let \(G\) be a compact connected simple Lie group of rank \(l\). Every finite-dimensional complex linear representation \(\rho : G \to \text{GL}(V)\) yields a character \(\chi = \text{tr}_ V \circ \rho\). If \(x \in G\) has finite order \(N\), then \(\chi(x)\) belongs to the cyclotomic field \(\mathbb{Q}(\zeta_ n)\) for every character \(\chi\), where \(\zeta_ N\) is a primitive \(N\)-th root of unity. The authors define \(A(x)\), the subfield of \(\mathbb{Q}(\zeta_ N)\) that is generated by all character values at \(x\). They concentrate on the case where \(N\) is a power of an odd prime, whence the extension \(\mathbb{Q}(\zeta_ N) /\mathbb{Q}\) is cyclic. In this case, \(A(x)\) is fully described by the degree of the extension \(\mathbb{Q}(\zeta_ N)/ A(x)\), which is called the depth \(b(x)\) of \(x\). Theorem 1 gives a polynomial formula for the number of conjugacy classes of elements of order \(N\) and depth \(b\). These polynomials may be expressed simply by means of the exponents of the Weyl group \(W\) of \(G\). For the proof, the authors employ a bijection between the set of conjugacy classes of elements of order \(N\) and certain Weyl group orbits. (Here \(W\) acts on the \(\mathbb{Z}\)-module \(\Gamma /N\Gamma\), where the lattice \(\Gamma\) is defined as the kernel of the exponential function for some maximal torus of \(G\).) The depth \(b(x)\) is then interpreted as the length of the orbit of the corresponding \(z \in \Gamma /N\Gamma\) under the stabilizer of the cyclic submodule \(\mathbb{Z} z\). Finally, Theorem 2 gives necessary and sufficient conditions for the existence of elements of depth \(b\) and order \(N = p^ k\).
Mathematics Subject Classification ID: 22C05
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 166880
zbMATH Keywords: compact connected simple Lie groups
zbMATH Keywords: finite-dimensional complex linear representations
zbMATH Keywords: cyclotomic fields
zbMATH Keywords: characters
zbMATH Keywords: character values
zbMATH Keywords: polynomial formula
zbMATH Keywords: number of conjugacy classes
zbMATH Keywords: exponents
zbMATH Keywords: Weyl groups
zbMATH Keywords: elements of order \(N\)
zbMATH Keywords: Weyl group orbits
zbMATH Keywords: lattices
zbMATH Keywords: exponential functions
zbMATH Keywords: depth
author: François Destrempes
author: Arturo Pianzola
reviewed by: Markus Stroppel
MaRDI profile type: Publication
cites work: Q3922829
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full work available at URL: https://doi.org/10.1007/bf01264522
OpenAlex ID: W1965753734
description: scientific article; zbMATH DE number 166880