While one key fact, about the trace of projection being the dimension of the subspace, was already mentioned, I think it is also important to mention another key fact about traces that is useful for characters. That is that the trace defines a natural inner product on the space of linear transforms.

Indeed, there is a natural inner product on $\operatorname{Hom}(V, V)$ defined in the coordinate form as

$$ \langle A, B \rangle = \sum\limits_{i, j} A_{ij} B_{ij}^*. $$

But more importantly, in the coordinate-agnostic form, this same inner product is expressed as

$$ \langle A, B \rangle = \sum\limits_{i} \left(\sum\limits_j A_{ij} B_{ij}^*\right) = \sum\limits_{i} (AB^*)_{ii} =\operatorname{tr} [AB^*]. $$

What's nice about it is that we can apply it to the representation $\rho : G \to \operatorname{Aut} V$, and get

$$ \langle \rho(g), \rho(h) \rangle = \operatorname{tr}\rho(gh^{-1}) = \chi(gh^{-1}). $$

Then, using the inner product on the regular representation, one may find that

$$ \frac{1}{|G|} \langle \rho(g), \rho(h) \rangle = \begin{cases} 1, & g=h, \\ 0, & g \neq h, \end{cases} $$

from which a lot of things derive, such as the Plancherel formula and the inverse Fourier transform.

While one key fact, about the trace of projection being the dimension of the subspace, was already mentioned, I think it is also important to mention another key fact about traces that is useful for characters. That is that the trace defines a natural inner product on the space of linear transforms.

Indeed, there is a natural inner product on $\operatorname{Hom}(V, V)$ defined in the coordinate form as

$$ \langle A, B \rangle = \sum\limits_{i, j} A_{ij} B_{ij}^*. $$

But more importantly, in the coordinate-agnostic form, this same inner product is expressed as

$$ \langle A, B \rangle = \sum\limits_{i} \left(\sum\limits_j A_{ij} B_{ij}^*\right) = \sum\limits_{i} (AB^*)_{ii} =\operatorname{tr} [AB^*]. $$

What's nice about it is that we can apply it to the representation $\rho : G \to \operatorname{Aut} V$, and get

$$ \langle \rho(g), \rho(h) \rangle = \operatorname{tr}\rho(gh^{-1}) = \chi(gh^{-1}). $$

Then, using the inner product on the regular representation, one may find that

$$ \frac{1}{|G|} \langle \rho(g), \rho(h) \rangle = \begin{cases} 1, & g=h, \\ 0, & g \neq h, \end{cases} $$

from which a lot of things can be derived, such as the Plancherel formula and the inverse Fourier transform.

While one key fact, about the trace of projection being the dimension of the subspace, was already mentioned, I think it is also important to mention another key fact about traces that is useful for characters. That is that the trace defines a natural inner product on the space of linear transforms.

Indeed, there is a natural inner product on $\operatorname{Hom}(V, V)$ defined in the coordinate form as

$$ \langle A, B \rangle = \sum\limits_{i, j} A_{ij} B_{ij}^*. $$

But more importantly, in the coordinate-agnostic form, this same inner product is expressed as

$$ \langle A, B \rangle = \sum\limits_{i} \left(\sum\limits_j A_{ij} B_{ij}^*\right) = \sum\limits_{i} (AB^*)_{ii} =\operatorname{tr} [AB^*]. $$

What's nice about it is that we can apply it to the representation $\rho : G \to \operatorname{Aut} V$, and get

$$ \langle \rho(g_i), \rho(g_j) \rangle = \operatorname{tr}\rho(g_i \circ g_j^{-1}) = \chi(g_i \circ g_j^{-1}). $$

Then, using the inner product on the regular representation, one may find that

$$ \frac{1}{|G|} \langle \rho(g_i), \rho(g_j) \rangle = \begin{cases} 1, & i=j, \\ 0, & i \neq j, \end{cases} $$

from which a lot of things derive, such as the Plancherel formula and the inverse Fourier transform.

While one key fact, about the trace of projection being the dimension of the subspace, was already mentioned, I think it is also important to mention another key fact about traces that is useful for characters. That is that the trace defines a natural inner product on the space of linear transforms.

Indeed, there is a natural inner product on $\operatorname{Hom}(V, V)$ defined in the coordinate form as

$$ \langle A, B \rangle = \sum\limits_{i, j} A_{ij} B_{ij}^*. $$

But more importantly, in the coordinate-agnostic form, this same inner product is expressed as

$$ \langle A, B \rangle = \sum\limits_{i} \left(\sum\limits_j A_{ij} B_{ij}^*\right) = \sum\limits_{i} (AB^*)_{ii} =\operatorname{tr} [AB^*]. $$

What's nice about it is that we can apply it to the representation $\rho : G \to \operatorname{Aut} V$, and get

$$ \langle \rho(g), \rho(h) \rangle = \operatorname{tr}\rho(gh^{-1}) = \chi(gh^{-1}). $$

Then, using the inner product on the regular representation, one may find that

$$ \frac{1}{|G|} \langle \rho(g), \rho(h) \rangle = \begin{cases} 1, & g=h, \\ 0, & g \neq h, \end{cases} $$

from which a lot of things derive, such as the Plancherel formula and the inverse Fourier transform.