Editing Representation theory of SU(2) (section)

===The characters===
The [[character (mathematics)|character]] of a representation <math>\Pi: G \rightarrow \operatorname{GL}(V)</math> is the function <math>\Chi: G \rightarrow \mathbb{C}</math> given by
:<math>\Chi(g) = \operatorname{trace}(\Pi(g))</math>.
Characters plays an important role in the [[Compact group#Representation theory of a connected compact Lie group|representation theory of compact groups]]. The character is easily seen to be a class function, that is, invariant under conjugation.

In the SU(2) case, the fact that the character is a class function means it is determined by its value on the [[maximal torus]] <math>T</math> consisting of the diagonal matrices in SU(2), since the elements are orthogonally diagonalizable with the spectral theorem.<ref>Travis Willse (https://math.stackexchange.com/users/155629/travis-willse), Conjugacy classes in $SU_2$, URL (version: 2021-01-10): https://math.stackexchange.com/q/967927</ref> Since the irreducible representation with highest weight <math>m</math> has weights <math>m, m - 2, \ldots, -(m - 2), -m</math>, it is easy to see that the associated character satisfies
:<math>\Chi\left(\begin{pmatrix}
  e^{i\theta} & 0\\
  0 & e^{-i\theta}
\end{pmatrix}\right) = e^{im\theta} + e^{i(m-2)\theta} + \cdots + e^{-i(m-2)\theta} + e^{-im\theta}.</math>

This expression is a finite geometric series that can be simplified to
:<math>\Chi\left(\begin{pmatrix}
  e^{i\theta} & 0\\
  0 & e^{-i\theta}
\end{pmatrix}\right) = \frac{\sin((m + 1)\theta)}{\sin(\theta)}.</math>

This last expression is just the statement of the [[Weyl character formula]] for the SU(2) case.<ref>{{harvnb|Hall|2015}} Example 12.23</ref>

Actually, following Weyl's original analysis of the representation theory of compact groups, one can classify the representations entirely from the group perspective, without using Lie algebra representations at all. In this approach, the Weyl character formula plays an essential part in the classification, along with the [[Peter–Weyl theorem]]. The SU(2) case of this story is described [[Compact group#The SU(2) case|here]].