For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda\_1,\lambda\_2,\ldots\lambda\_m$ with probability distribution
$$P\_\beta(\lambda\_1,\lambda\_2,\ldots\lambda\_m)\propto \prod\_{k=1}^{m}\lambda\_k^{\beta m/2}e^{-\beta\lambda\_k/2}\prod\_{1\leq i<j\leq m}|\lambda\_i-\lambda\_j|^\beta,\;\;\lambda\_k\geq 0.\qquad [1]$$
The index $\beta\in\{1,2,4\}$.


An efficient way to generate these random variables, discussed for example in [Forrester's book,](http://books.google.nl/books?id=C7z3NgOlb1gC&q=wishart+distribution&hl=en) is to use the eigenvalues $\mu\_1,\mu\_2,\ldots\mu\_m$ of Wishart matrices $W^\dagger W$, constructed from a real ($\beta=1$), complex ($\beta=2$) or quaternion ($\beta=4$) matrix $W$ of dimension $n\times m$ (with $n\geq m$). If the matrix elements are i.i.d. from a normal distribution, the eigenvalue distribution is


$$P\_\beta(\mu\_1,\mu\_2,\ldots\mu\_m)\propto \prod\_{k=1}^{m}\mu\_k^{\beta a/2}e^{-\beta\mu\_k/2}\prod\_{1\leq i<j\leq m}|\mu\_i-\mu\_j|^\beta,\;\;\mu\_k\geq 0,\qquad [2]$$


with $a=n-m+1-2/\beta$.


I can identify the distributions [1] and [2] by equating $a=m\Rightarrow n=2m-1+2/\beta$. This works for $\beta=1$ and for $\beta=2$, but fails for $\beta=4$, because it would imply a half-integer dimension of $W$.


**Question:** Is there a work-around which would allow me to generate the distribution [1] for $\beta=4$ using Wishart matrices? I'm basically asking for a way to make sense of "half a quaternion" --- perhaps I'm asking too much and the answer is just "no way".



Quaternion Wishart matrices of half-integer dimension?
This is not an answer to trying to define a half quaternion; instead,the construction of
a random matrix model for arbitrary $a> \beta(m-1)/2$ and arbitrary $\beta> 0$ by using tri-diagonal matrices (generalizing Wishart distributions) was 
done by Dumitiru and Edelman:
<http://www.math.washington.edu/~dumitriu/JMathPhys_43_5830.pdf>


