# 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 $V$.

# 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 $\mathbb{Q}$.

# Fundamental theorem of algebra…?

Let $C_{n}$ be the cyclic group of order $n$ and consider $\mathbb{C}C_n$ the group algebra. Prove that the number of solutions to the equation $x^k = x$ for $x\in\mathbb{C}C_n$ is $k^n$ for $k>1$.

# Infinite dimensional Schur

Let $A$ be an algebra over $\mathbb{C}$ and let $V$ be an irreducible representation of $A$ with at most countable basis. Then any homomorphism of representations $\phi:V\to V$ is a scalar operator.

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# An indecomposable module

Suppose $A=k[x_1,\dots,x_n]$ and let $I$ be a proper ideal containing all monomials of degree $\geq N$. Then $A/I$ is an indecomposable $A$-module.

# 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. Suppose $\varphi\in D$ 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.