Editing Representation theory of SU(2) (section)

==The group representations==
===Action on polynomials===
Since SU(2) is simply connected, a general result shows that every representation of its (complexified) Lie algebra gives rise to a representation of SU(2) itself. It is desirable, however, to give an explicit realization of the representations at the group level. The group representations can be realized on spaces of polynomials in two complex variables.<ref>{{harvnb|Hall|2015}} Section 4.2</ref> That is, for each non-negative integer <math>m</math>, we let <math>V_m</math> denote the space of homogeneous polynomials <math>p</math> of degree <math>m</math> in two complex variables. Then the dimension of <math>V_m</math> is <math>m + 1</math>. There is a natural action of SU(2) on each <math>V_m</math>, given by
:<math>[U \cdot p](z) = p\left(U^{-1}z\right),\quad z\in\mathbb C^2, U\in\mathrm{SU}(2)</math>.

The associated Lie algebra representation is simply the one described in the previous section. (See [[Representation theory of semisimple Lie algebras#The case of sl(2,C)|here]] for an explicit formula for the action of the Lie algebra on the space of polynomials.)

===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]].

===Relation to the representations of SO(3)===
{{see also|Rotation group SO(3)#Connection between SO(3) and SU(2)|Projective representation}}
Note that either all of the weights of the representation are even (if <math>m</math> is even) or all of the weights are odd (if <math>m</math> is odd). In physical terms, this distinction is important: The representations with even weights correspond to ordinary representations of the [[rotation group SO(3)]].<ref>{{harvnb|Hall|2015}} Section 4.7</ref> By contrast, the representations with odd weights correspond to double-valued (spinorial) representation of SO(3), also known as [[projective representation]]s.

In the physics conventions, <math>m</math> being even corresponds to <math>l</math> being an integer while <math>m</math> being odd corresponds to <math>l</math> being a half-integer. These two cases are described as [[integer spin]] and [[half-integer spin]], respectively. The representations with odd, positive values of <math>m</math> are faithful representations of SU(2), while the representations of SU(2) with non-negative, even <math>m</math> are not faithful.<ref>{{Cite book|url=https://books.google.com/books?id=1jw8DQAAQBAJ|title=Group Theory for Physicists|last=Ma|first=Zhong-Qi|date=2007-11-28|publisher=World Scientific Publishing Company|isbn=9789813101487|pages=120|language=en}}</ref>