What is an irreducible character of a finite group?

Solution 1:

The homomorphisms $G\to \Bbb C^\times$ are actually not the whole story of character theory, but are a very tidy chapter in it. If $V$ is a vector space (over $\Bbb C$) and $G$ finite, the homomorphisms $G\to GL(V)$ from $G$ into the general linear group of invertible linear maps are called representations, which are essentially the ways to equip $V$ with a linear $G$ action. If $\rho$ is a representation, then the map given by $\chi_\rho:G\to \Bbb C:g\mapsto\mathrm{tr}\,\rho(g)$ (the trace of the linear map associated to $g$, which is independent of basis or coordinate choice for $V$) is called a character of $G$.

If $V$ is one-dimensional (in which case we call $\rho$ and $\chi_\rho$ one-dimensional as well) then $\rho=\chi_\rho$ and the characters are multiplicative. Note that $\mathrm{tr}\,\rho(e_G)=\dim\,V$ shows the dimension can be directly computed from the character, so there is no ambiguity with respect to what dimension a character may have. With a distinguished basis we have $V\cong \Bbb C^n$ in an obvious way, and so we can write $GL(V)$ as $GL_n(\Bbb C)$, in which case we are working with matrix representations specifically.

If there is a proper nontrivial subspace $W\subseteq V$ that is invariant under the linear $G$-action, that is if we have $\rho(g)W\subseteq W$ for each $g\in G$, then the restriction of $G$'s linear action to $W$ then forms a subrepresentation of $V$. If a representation $\rho:G\to GL(V)$ has no subrepresentations, it is called an irreducible representation, in which case $\chi_\rho$ is also called irreducible. This is well-defined because (over any field of characteristic zero) representations are uniquely determined by their characters, which again prevents any sort of ambiguity or conflicting descriptions from arising.

A one-dimensional space has no proper nontrivial subspaces, so one-dimensional representations and characters (the ones you are talking about) are all automatically irreducible. Using the very nifty Schur's lemma, one can see that representations of abelian groups decompose into a direct sum of one-dimensional representations and thus are irreducible precisely when they are one-dimensional (you may encounter Schur's and direct sums later).

I can recommend this excellent note on the representation theory of the symmetric groups and Young tableaux, where the irreducible representations are obtained through Specht modules. To understand this fully you will need relatively good familiarity with representation theory, algebra, combinatorics and permutations, so maybe come back to it later. (Also for other readers.)

Representation theory doesn't need to be done in $\Bbb C$, we can work in arbitrary fields. However, as I hinted at a moment ago, if the characteristic divides the order of the finite group $G$ then things get complicated and we are working in modular representation theory. Things also get complicated when $G$ is a topological group, in which case we need theory from abstract harmonic analysis.

Solution 2:

There are two somewhat conflicting definitions of characters, and the one you cited is not the relevant one.

1. A character $\chi$ of a linear representation $\pi$ of a group $G$ is its composition with trace $\operatorname{tr}\circ \pi$.
2. A character of a group is a character of any its representation.
3. An irreducible character is the character of an irreducible representation.

What you cited is a multiplicative character. (Over $\bf C$, not sure how it is in general), multiplicative characters are exactly the characters of one-dimensional representations, and they are irreducible.

So a multiplicative character is an irreducible character, but not the other way around: any $S_n$ has a natural irreducible $n-1$-dimensional representation.

Worth noting, iirc, the only multiplicative characters of a symmetric group are the trivial one and the sign, so they are rather irrelevant in this case.

On the other hand, for abelian groups, irreducible characters are exactly the one-dimensional (so multiplicative) characters (that's why in the context of abelian groups, commutative Fourier analysis etc., the definition you cited is usually used).