Family of $n!+1$ non-commuting matrices in unitary group

Question

I am looking to find $n!+1$ matrices in the unitary group $U(n)$ which do not pairwise commute. I believe that the $n!$ matrices given by starting with the $n\times n$ matrix $$U:=\begin{pmatrix}-1&0\\0&\mathrm{id}_{n-1}\end{pmatrix},$$ then permuting its rows, will not actually commute. Is this correct? If so, what matrix can we add to this family to keep the “pairwise non-commuting” property?

Answer 1

No, your construction doesn't work. $\pmatrix{-1\\ &0&1\\ &1&0\\ &&&1&0\\ &&&0&1}$ and $\pmatrix{-1\\ &1&0\\ &0&1\\ &&&0&1\\ &&&1&0}$ commute, for instance.

When $n\ge3$, it is actually very easy to produce an uncountably infinite set of non-commuting unitary matrices: for each different line that passes through the origin in $\mathbb R^3$, pick a rotation matrix $R$ for some angle in $(0,\pi)$ about that line. Then take $\pmatrix{R\\ &I_{n-3}}$ as the unitary matrix.