Are there involutory unitary matrices U and V such that the group generated by U and V is isomorphic to $\mathbb{Z}/2 \ast \mathbb{Z}/2$? If so, how many such pairs of matrices are there? Is there a known way to classify or at least generate examples of such pairs? Alternatively, can you prove no such pairs exist?
I am mostly interested in the case where U and V are $N \times N$ matrices with $N = 2^n$ for positive integers $n$, but would also be interested in any special cases.
Thoughts so far: If U and V are involutions and unitary, then each must be Hermitian also. Thus, each of U and V must be a matrix with all eigenvalues equal to $\pm 1$.
Also, since $\langle U, V \rangle \cong \mathbb{Z}/2 \ast \mathbb{Z}/2$ and since U and V are involutions, $UVUV = UVU^{-1}V^{-1} = [U,V] \neq I$, so U and V cannot commute, and in fact they also cannot anticommute since that would mean $[U,V]^2 = I$, which would again fail our requirements.
Another line of thinking comes from an answer to this question (Matrices which are both unitary and Hermitian) which states that a matrix U is Hermitian and Unitary if and only if $U = 2P – I$, for some orthogonal projection $P$. I'm not sure if this helps.
Any ideas?
Best Answer
Almost all pairs of involutions satisfy this property.
Unitary involutions are in correspondence with subspaces, so the space of pairs $(U,V)$ (or representations of $\mathbb Z/2 *\mathbb Z/2$) is a disjoint union of products of Grassmannians, of course.
The condition that $U,V$ determine a faithful representation of $\mathbb Z/2*\mathbb Z/2$ is equivalent to $\langle UV\rangle \cong \mathbb Z$. Another way to see this condition is to remove the pairs $(U,V)$ such that $(UV)^n = I$ for each $n$. Since the matrix equation is just a set of polynomial equations in the coefficients of the matrices, for fixed $n$ it defines a subvariety. On the $G_\delta$ set away from the countable union of these subvarieties, the condition is true. This is what algebraic geometers call a "very generic property."
We're over an uncountable base field, so a very generic property always holds on a non-empty set. In more analytic language, a subvariety has measure 0, so by subadditivity their countable union also has measure zero. Thus the property holds almost everywhere.