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 .
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 .
Let be the cyclic group of order and consider the group algebra. Prove that the number of solutions to the equation for is for .
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.
Suppose and let be a proper ideal containing all monomials of degree . Then is an indecomposable -module.
(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 .
- By Schur’s lemma, is a division algebra. Moreover, it has a countable basis.
- Suppose does not act as a scalar. Then is trascendental over .
- This is absurd since it would imply has uncountable dimension.