Why does the trivial representation have degree 1?

If you have a representation from $G \to Aut(V)$, it has degree $1$ if $V$ is a 1-dimensional vector space over $F$. The trivial representation sends any element of $G$ to the trivial automorphism $v \mapsto v$, but I don't get why this means $V$ has dimension $1$. Help?

Thanks!

Solution 1:

It is necessary to distinguish between a trivial representation and the trivial representation.

The trivial representation is by definition the one dimensional representation that sends every $g\in G$ to the identity. This is, up to isomorphism, the only irreducible trivial representation.

In general, if you have some other representation $\rho$ with $\rho(g)=\operatorname{Id}_V$ for all $g\in G$ (which I am fine with calling "a trivial representation), then every subspace of $V$ will be invariant, and so taking any basis, we split $V$ into a direct sum of one dimensional representations, each one isomorphic to the trivial representation.

Generally speaking, at least in nice situations, we only have names for the irreducible representations, because in nice situations, everything is built up out of these representations in simple ways.

For example, $S_n$ naturally acts on $\mathbb{C}^n$, and while we could give a name to this representation, we observe that the vector space spanned by $(1,1,1,1\ldots, 1)$ is a copy of the trivial representation, and the complement, $\{(a_1, \ldots a_i) | \sum a_i=0 \}$ will be an irreducible representation, called the standard representation. All the other irreducible representation of $S_n$ have "names" given by partitions of $n$ (the definition is complicated, and can be found here).

The one exception to this is the regular representation, which is what you get when you take a basis element $e_g$ for every $g\in G$ and define an action by $g'e_g=e_{g'g}$. This representation is not irreducible, and (for finite groups and representations over $\mathbb{C}$) contains every irreducible representation with multiplicity the dimension of the representation.