The probability distribution of spectral moments for the Gaussian beta-ensembles

We derive the joint probability distribution of the first two spectral moments for the G\(\beta\)E random matrix ensembles in N dimensions for any N. This is achieved by making use of two complementary invariants of the domain in \(\mathbb{R}^N\) where the spectral moments are defined. Our approach is significantly different from those employed previously to answer related questions and potentially offers new insights. We also discuss the problems faced when attempting to include higher spectral moments.