Consider the space $\mathrm{SU}(2)^\natural$ of conjugacy classes in $\mathrm{SU}(2)$. It has a natural identification with the interval $[0,\pi]$ with Haar measure $\frac{2}{\pi} \sin^2\theta\, \mathrm{d}\theta$, via the mapping 
$$
\theta \mapsto x\_\theta = \begin{pmatrix} e^{i\theta} \\ & e^{-i\theta}\end{pmatrix} .
$$
Under this identification, the map $U\_k\colon \mathrm{SU}(2)^\natural \to \mathbf{R}$ given by $U\_k(x) = \mathrm{tr}(\mathrm{sym}^k x)$ is 
$$
U\_k(\theta) = \frac{\sin((k+1)\theta)}{\sin(\theta)} .
$$



> 
> I am wondering: is the exact value of the total variation of the functions $U\_k$ known? In this context, this means: is there an exact formula (or asymptotic for)
>  $$
> \int\_0^\pi |U\_k'(\theta)|\, \mathrm{d}\theta ?
> $$
> 
> 
> 


More generally, for $G$ a compact Lie group, we can identify $G^\natural$ with a quotient of Euclidean space via $\exp\colon \mathfrak{t} \to G^\natural$ for $\mathfrak{t}$ the Lie algebra of a maximal torus. Is there any reasonable formula for the total variation of the trace of an irreducible representation of $G$, in terms of the corresponding highest weight? 



Variation of trace of symmetric powers
For the large-$k$ asymptotics I would first approximate $|U'\_k(\theta)|$ by its envelope
$$F\_k(\theta)=\frac{k}{\sqrt{2}\sin\theta},$$
plotted together for $k=50$:


![](https://ilorentz.org/beenakker/MO/envelope.png)


The divergence of $F\_k(\theta)$ at $\theta=0$ is cut-off at $\theta\_1=1/k$, and similarly the divergence at $\theta=\pi$ is cut-off at $\theta\_2=\pi-1/k$. Integration of $F\_k(\theta)$ from $\theta\_1$ to $\theta\_2$ produces the asymptotics


$$\int\_{0}^\pi |U'\_k(\theta)|\,d\theta\rightarrow\sqrt{2}\,k\log(2k),\;\;k\gg 1.$$


A numerical evaluation of the integral (orange points) seems quite close to this asymptote (blue points):


![](https://ilorentz.org/beenakker/MO/asymptotic_2.png)


