Representation Theory

A representation of a group $G$ on a vector space $V$ over a field $k$ is a group homomorphism \[ \rho : G \to GL(V). \] The dimension $\dim_k V$ is called the degree of the representation.

A subspace $W \subseteq V$ is a subrepresentation (or $G$-invariant subspace) if $\rho(g)(W) \subseteq W$ for all $g \in G$. This notion relies on the setup of .

A representation $(\rho, V)$ (in the sense of ) is irreducible (or simple) if $V \neq 0$ and the only subrepresentations (see ) of $V$ are $\{0\}$ and $V$ itself.

Let $(\rho, V)$ and $(\sigma, W)$ be irreducible representations () of $G$ over an algebraically closed field $k$. • Any $G$-module homomorphism $\phi : V \to W$ is either zero or an isomorphism. • $\mathrm…

Let $G$ be a finite group and $k$ a field with $\mathrm{char}(k) \nmid |G|$. Then every representation () of $G$ over $k$ is completely reducible, i.e.\ decomposes into irreducibles ().

The regular representation of $G$ is the action of $G$ on the group algebra $k[G]$ by left multiplication (a special case of ): \[ \rho(g) \cdot \sum_{h \in G} a_h \, h \;=\; \sum_{h \in G} a_h \, (gh). \] By , over a…

Over $\mathbb{C}$, the number of (isomorphism classes of) irreducible representations of $G$ equals the number of conjugacy classes of $G$. This follows from and .

For a finite group $G$ over $\mathbb{C}$, combining and , \[ \sum_{\chi \in \hat{G}} (\dim V_\chi)^2 = |G|. \]

Using and , compute the complete character table of the symmetric group $S_3$. Verify the column orthogonality relations: \[ \sum_{\chi \in \hat{G}} \chi(g)\,\overline{\chi(h)} \;=\; |C_G(g)| \cdot \delta_{[g],[h]}…

The category of $kG$-modules and the category of representations over a group $G$ $Rep_k (G)$ are equivalent.