Let be a finite group (or a compact Lie group). Prove that if is a faithful finite dimensional complex representation of then any irreducible representation embeds in some tensor product of .

# representation theory

# Be rational

Most of representation theory results work out for algebraically closed fields. For instance, we can classify irreducible representations for the dihedral group with elements over . Classify all irreducible representations for over .

# Fundamental theorem of algebra…?

Let be the cyclic group of order and consider the group algebra. Prove that the number of solutions to the equation for is for .

# Infinite dimensional Schur

Let be an algebra over and let be an irreducible representation of with at most countable basis. Then any homomorphism of representations is a scalar operator.

# Deranged cohomology

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# An indecomposable module

Suppose and let be a proper ideal containing all monomials of degree . Then is an indecomposable -module.

# A generalization of Schur’s lemma

(Dixmier) Let be a -algebra (in fact, any uncountable algebraically closed field will do) and be a simple -module having a countable basis as a -vector space. Then .

Brief outline:

- By Schur’s lemma, is a division algebra. Moreover, it has a countable basis.
- SupposeĀ does not act as a scalar. Then is trascendental over .
- This is absurd since it would imply has uncountable dimension.