Calculating Mass of a Subset in SO(n) Using Definite Integrals

[jimbo11]

I'm wrestling with this problem and I'd appreciate any pointers anyone can give me. (I've tried Google, but no luck.)

I'm trying to figure out the total mass, with respect to the Haar measure I guess, that's accounted for by a definite subset of \mathbf{SO}(n). Specifically, working in \mathbb{R}^n, we're given a minimum and maximum rotation between each pair of coordinates: say \theta_{12} \in \left[ \underline{a}, \overline{a} \right], \theta_{13} \in \left[ \underline{b}, \overline{b} \right] and so on. If we call the rectangle bounded by those limits \Theta, I need to work out \int_\Theta dO(\vec{\theta}).

I'm a novice at this, so any suggestions would be gratefully received!

 

[henry_m]

I'm not quite sure what the exact subset you need to find the size of is. But the general strategy should be as follows:

i) Find some convenient parameters for the group (or part of it).
ii) Find the Haar measure in terms of those parameters.
iii) Identify the region of integration in terms of these parameters.
iv) Integrate!

The choice in (i) will probably be mostly motivated by trying to make the form of the region in (iii) simple.

 

[jimbo11]

Fair enough. Thanks!