Determining representations from invariant dimensions
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.
scientific article; zbMATH DE number 4125714