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A random matrix decimation procedure relating $β= 2/(r+1)$ to $β= 2(r+1)$

Classical random matrix ensembles with orthogonal symmetry have the property that the joint distribution of every second eigenvalue is equal to that of a classical random matrix ensemble with symplectic symmetry. These results are shown to be the case $r=1$ of a family of inter-relations between eigenvalue probability density functions for generalizations of the classical random matrix ensembles referred to as $β$-ensembles. The inter-relations give that the joint distribution of every $(r+1)$-st eigenvalue in certain $β$-ensembles with $β= 2/(r+1)$ is equal to that of another $β$-ensemble with $β= 2(r+1)$. The proof requires generalizing a conditional probability density function due to Dixon and Anderson.