Confusion about Representation theory and Characters

Question

I just started Representation theory and am quite confused by the notation and the form of representations.

Firstly the definition of a representation we were given was.

"A representation of a group $G$ is a finite-dimensional vector space V over $\mathbb{C}$ equipped with a group morphism $$\rho :G\rightarrow \text{Aut}_{\mathbb{C}}(V)$$"

So each element in G gets mapped to an invertible linear transformation $T_g :V \rightarrow V$ ? So is our vector space a space of invertible matrices? (In finite case anyway)

Secondly I'm confused about character of a representation.

The definition given,

"the character of $(V,\ \rho)\in\ \text{Rep}(G)$ is the map $$\chi ^{(V,\ \rho )} :G \rightarrow \mathbb{C}\\ g\mapsto \text{tr}(\rho (g)$$

So here the character is defined in terms of a single element, but does our representation not have a matrix, defined by $\rho (g)$ for each $g \in G$? how does this give the character of the entire representation if it's defined in terms of a single element?

Obviously I have fundamental misconceptions of this subject and the objects involved, I've only had three lectures so far so hoping to clear them up and keep learning.

If anyone could help me get a clearer idea of the objects involved I'd really appreciate it!

Answer 1

Perhaps the thing to do here is to unravel the definition. If $V$ is a complex vector space, then $Aut(V)$ is the space of $C$-linear invertible maps $V\rightarrow V$. Fix a basis of $V$. Then a linear map $f:V\rightarrow V$ is uniquely represented by a some $\dim V\times \dim V$ matrix $A$ (with respect to the basis of $V$) with entries in $C$ in the sense that $f(v)=Av$ for all $v\in V$. This gives an isomorphism of $Aut(V)$ (a group) with $GL_N(C)$, where $N=\dim V$. So a representation $\rho:G\rightarrow Aut(V)$ can be reinterpreted as a homomorphism $\rho:G\rightarrow GL_N(C)$. This is obviously dependent on the choice of basis, but since any two bases of $V$ are conjugate, this isn't a problem.

In short: a representation is a way of assigning to each $g\in G$ a linear endomorphism of $V$ or, equivalently, an $N\times N$ matrix with respect to some fixed choice of basis.

The character $\chi_\rho$ of $\rho:G\rightarrow Aut(V)$ is defined by $\chi_\rho(g)=\mathrm{tr}(\rho(g))$; this is simply the trace of $\rho(g)$ as a linear operator, which is the sum of the diagonal elements, or the sum of the eigenvalues, of the matrix representation of $\rho(g)$ with respect to any choice of basis.

Here's one last way of rephrasing the above: a representation of $G$ is a pair $(\rho,V)$, where $V$ is a $C$-vector space and $\rho:G\rightarrow Aut(V)$ is a homomorphism. This is precisely the same as saying that a representation is a group action of $G$ on some $C$-vector space.

A completely trivial example to illustrate what's going on: write $Z/2Z=\{1,g\}$, where $g^2=1$ is the non-trivial element. Then a 1-dimensional representation of $Z/2Z$ is defined by $1\mapsto 1$, $g\mapsto -1$. This is $Z/2Z$ acting on the 1-dimensional $C$-vector space, which I've identified with $C$ by choosing the basis $\{1\}$. The group $Aut(V)$ is equal to $C^\times=GL_1(C)$.

If that's still unclear, try to work out an example in higher dimensions. For example, you can represent $Z/NZ$ on a 2-dimensional vector space by identifying $Z/NZ$ with $\{0,\dots,N-1\}$ modulo $N$, and letting $k$ act as rotation by $e^{2\pi i k/N}$ (which is a representable by a $2\times 2$ orthogonal matrix). (And if you want another example after that, let $S_n$ act on $C^n$ w.r.t some basis $\{v_1,\dots,v_n\}$ in the following way: if $\sigma\in S_n$ is a permutation of $\{1,\dots,n\}$, then define $\rho:S_n\rightarrow Aut(C^n)$ by $\rho(\sigma)(a_1v_1+\dots+a_nv_n)=a_1v_{\sigma(1)}+\dots+a_nv_{\sigma(n)}$.)

Answer 2

The mistake you seem to be making is that the character is somehow determined by a single element of $G$. This is wrong.

A character is a function $$\chi_\rho:G\to\mathbb{C}$$ given by $\chi_\rho(g)=\mathrm{tr}(\rho(g))$ for each $g\in G$.

Example: The left regular representation of $G$ is defined as follows: let $V$ be a vector space with basis $\{v_g\mid G\in G\}$, and define $$\rho:G\to \mathrm{Aut}(V)$$ by $\rho(g)(v_h)=v_{gh}$. In this case elements of $G$ are represented by permutation matrices, and $$\chi_\rho(g)=\begin{cases}|G|&\mbox{if }g=1\\0&\mbox{otherwise.}\end{cases}$$

Example: The group $S_3$ has a 2-dimensional representation $\rho:S_3\to GL_2(\mathbb{C})$ given by \begin{align} \rho(1)&=\begin{pmatrix}1&0\\0&1\end{pmatrix}\\ \rho((12))&=\begin{pmatrix}-1&1\\0&1\end{pmatrix}\\ \rho((23))&=\begin{pmatrix}1&0\\1&-1\end{pmatrix}\\ \rho((123))&=\begin{pmatrix}0&-1\\1&-1\end{pmatrix}\\ \rho((321))&=\begin{pmatrix}-1&1\\-1&0\end{pmatrix}\\ \rho((13))&=\begin{pmatrix}0&-1\\-1&0\end{pmatrix}\\ \end{align} Therefore, the character is given by $$\chi_\rho(g)=\begin{cases}2&\mbox{if }g=1\\ 0&\mbox{if }g=(ij)\mbox{ is a transposition}\\ -1&\mbox{if }g=(ijk)\mbox{ is a 3-cycle.}\end{cases}$$