Faithful representations and tensor products

Let G be a finite group (or a compact Lie group). Prove that if \rho:G\to\mathrm{GL}(V) is a faithful finite dimensional complex representation of G then any irreducible representation embeds in some tensor product of V.

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Faithful representations and tensor products

Be rational

Most of representation theory results work out for algebraically closed fields. For instance, we can classify irreducible representations for the dihedral group D_{2n} with 2n elements over \mathbb{C}. Classify all irreducible representations for D_{2n} over \mathbb{Q}.

Be rational

A generalization of Schur’s lemma

(Dixmier) Let A be a \Bbb{C}-algebra (in fact, any uncountable algebraically closed field will do) and V be a simple A-module having a countable basis as a \Bbb{C}-vector space. Then D=\mathrm{End}_A(V)\simeq \Bbb{C}.

Brief outline:

  1. By Schur’s lemma, D is a division algebra. Moreover, it has a countable basis.
  2. SupposeĀ \varphi\in D does not act as a scalar. Then \varphi is trascendental over \Bbb{C}.
  3. This is absurd since it would imply \Bbb{C}(\varphi)\subseteq D has uncountable dimension.
A generalization of Schur’s lemma