5
votes
1answer
189 views

Haar measure on $O(n)$ reduced to simpler probability space

The background of this question is how a random variable $X$ on the orthogonal group $O(n)$ whose distribution is the normalized Haar measure $\mu$, i.e., $\mu( O(n) ) = 1$, can be realized on a ...
2
votes
1answer
300 views

Is every subgroup of a connected unimodular (matrix) Lie group also unimodular?

My intuition is that the answer is yes: Let $G$ be the original group, and let $H$ be a subgroup of $G$. Let $\mu$ be a Haar measure on $G$ that is both right- and left-invariant. I think that if we ...
4
votes
1answer
360 views

Decomposition of Haar measure other than Hurwitz's

Hurwitz defined a decomposition of the Haar measure on $SO(n)$ based on Given's rotation. So by left multiplication of Givens rotation one can always bring an orthogonal matrix into the identity. The ...
10
votes
2answers
1k views

Volume of fundamental domain and Haar measure

In my research, I do need to know the Haar measure. I have spent some time on this subject, understanding theoretical part of the Haar measure, i.e existence and uniqueness, Haar measure on quotient. ...