Are there Hermitian Unitary matrices U and V generating $\mathbb{Z}/2 \ast \mathbb{Z}/2$?

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?

• You could just take $U$ and $V$ to be two real orthogonal reflections about lines through the origin such that the angle between them is not a rational multiple of $\pi$. Then $UV$ has infinite order. – Derek Holt Nov 29 '18 at 9:22

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."
• Are you taking the correspondence between unitary involutions and subspaces to be via the $U = 2P - I$ decomposition I mentioned above? – wanderingmathematician Nov 30 '18 at 15:52
• @user334137 Yes, $U$ corresponds to the image of $P$. – Ben Nov 30 '18 at 15:56