I have a conjecture that I am trying to either prove or disprove, so far without much luck.
Suppose that $\mathbf{R}$ is a random square upper triangular matrix over $GF(2)$, which is invertible (i.e. the diagonal has all-ones, a necessary condition for invertibility clearly), and the off-diagonal elements are selected as Ber($\frac{1}{2}$) random variables.
What is the distribution of $\mathbf{R}^{-1}$ (where the inverse is over GF(2))? My conjecture is that $\mathbf{R}^{-1}$ is also a square random upper triangular matrix whose off-diagonal elements are all Ber($\frac{1}{2}$) (easy to prove) r.v.'s, AND mutually independent (difficult to prove).
For small examples such as for dimensions $n=2,3,4,5$ I can indeed check that the above is true. Any ideas on how to prove this?
Selecting the off-diagonal elements as independent Bernoulli variables with probability $\frac12$ leads to a uniform distribution over all invertible upper triangular $n\times n$ matrices. These are in bijection with their inverses, so taking the inverse leaves this uniform distribution invariant. Since the inverses are again uniformly distributed, their off-diagonal elements are again independent Bernoulli variables with probability $\frac12$.