Linear representation theory of group of unit quaternions

This article gives specific information, namely, linear representation theory, about a particular group, namely: group of unit quaternions.
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This page concerns the linear representation theory of the group of unit quaternions, denoted $S^{3},SU(2),S^{0}(\mathbb {H} ),S^{1}(\mathbb {C} )$ .

On this page we shall consider this group as the special unitary group $SU(2)$ , that is the set of matrices of the form $\{A\in GL_{2}(\mathbb {C} ):A{\overline {A^{T}}}=I\}$ under matrix multiplication.

Summary

Item	Value
Degrees of irreducible representations over a splitting field (such as $\mathbb {C}$ )	1,2,3,4,5,... (all positive integers)

Complex representations

Let $V_{n}$ be the complex vector space of homogenous polynomials in two variables of degree $n$ .

$V_{n}$ has dimension $n+1$ since $V_{n}=\sum _{i=0}^{n}\mathbb {C} x^{i}y^{n-i}$ .

Then note $SU(2)$ acts on $V_{n}$ via the map $p_{n}$ defined by

$p_{n}\left({\begin{pmatrix}a&b\\c&d\end{pmatrix}}\right)x^{i}y^{j}=(ax+cy)^{i}(bx+dy)^{j}$ .

It turns out that these are precisely the irreducible complex representations of $SU(2)$ . (EDITOR NOTE: MAKE PROOF PAGE PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

)

In particular, note, $V_{0}$ is the trivial representation.

Characters

Note that for any $A\in SU(2)$ , $A$ is conjugate to some element of the subgroup

$T:=\left\{{\begin{pmatrix}z&0\\0&z^{-1}\end{pmatrix}}:z\in \mathbb {C} ,|z|=1\right\}\leq SU(2)$ .

Then,

$p_{n}\left({\begin{pmatrix}z&0\\0&z^{-1}\end{pmatrix}}\right)(x^{i}y^{j})=(zx)^{i}(z^{-1}y)^{j}=z^{i-j}x^{i}y^{j}$ .

Thus giving

$\chi _{V_{n}}\left({\begin{pmatrix}z&0\\0&z^{-1}\end{pmatrix}}\right)=z^{n}+z^{n-2}+\dots +z^{-(n-2)}+z^{-n}={\frac {z^{n+1}-z^{-(n+1)}}{z-z^{-1}}}$ .

Since every element of the group is conjugate to something in $T$ , and characters are class functions, this defines the character on every element of the group.

Haar integral and measure

Haar integral for class functions

The Haar integral on $SU(2)$ on class functions is simple, and can be computed as follows.

Recall that every $A\in SU(2)$ is conjugate to a matrix in the subgroup $T:=\left\{{\begin{pmatrix}z&0\\0&z^{-1}\end{pmatrix}}:z\in \mathbb {C} ,|z|=1\right\}\leq SU(2)$ .

Given a matrix $A\in SU(2)$ , denote $z(A)\in \mathbb {C}$ to be such that $A$ is conjugate to ${\begin{pmatrix}z(A)&0\\0&z(A)^{-1}\end{pmatrix}}$ .

Then, if $f$ is a class function,

$\int _{SU(2)}f(A)dA={\frac {1}{\pi }}\int _{0}^{2\pi }f(z(A))\sin ^{2}(z(A))d\theta$ , where $z(A)=e^{i\theta }$ .

In particular, this is useful for calculating orthogonality of characters of $SU(2)$ , which is a useful way to decompose a representation of the group into irreducibles.