Determining representations from invariant dimensions
type: Item
instance of: scholarly article
title: Determining representations from invariant dimensions
zbMATH Open document ID: 0687.22004
DOI: 10.1007/BF01233432
published in: Inventiones Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143839
review text: Let \(\rho\) : \(G\hookrightarrow GL_ n\) be a faithful representation of a connected semisimple complex Lie group. ``Dimension data'' for the pair (G,\(\rho)\) is the data associating dim \(W^ G\) to every representation \(GL_ n\to GL(W)\). The authors prove: Theorem 1. Dimension data uniquely determines G up to isomorphism. Theorem 2. If \(\rho\) is irreducible, dimension data uniquely determines \(\rho\) up to isomorphism. Theorem 3. In the full generality of Theorem 1, \(\rho\) is not determined up to isomorphism by dimension data. In any case dimension data determines \(\rho (T)\subset GL_ n\) up to conjugation, where T is a maximal torus of G ({\S}1 Proposition 1). In other words, the weight configuration of \(\rho\) is determined. In a separate approach the authors take this information, instead of dimension data, as a starting point and prove Theorem 4 which completely determines all nonisomorphic pairs (G,\(\rho)\), with \(\rho\) irreducible, having the same weight configuration.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 4125714
zbMATH Keywords: faithful representation
zbMATH Keywords: connected semisimple complex Lie group
zbMATH Keywords: dimension data
zbMATH Keywords: maximal torus
zbMATH Keywords: weight configuration
author: Michael Larsen
author: Richard Pink
MaRDI profile type: Publication
cites work: Q3941610
cites work: Semisimple subalgebras of semisimple Lie algebras
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Q3894074
cites work: Q5772226
OpenAlex ID: W2088627758
description: scientific article; zbMATH DE number 4125714