Products of reflections in the unitary group

Abstract:

Let $A \in U(n),\det (A) = \pm 1$ and let $\exp (i{\alpha _k}),1 \leqslant k \leqslant n$ be the eigenvalues of A where $0 \leqslant {\alpha _1} \leqslant {\alpha _2} \leqslant \cdots \leqslant {\alpha _n} < 2\pi$. Then $k(A) = ({\alpha _1} + \cdots + {\alpha _n})/\pi$ is an integer and $0 \leqslant k(A) \leqslant 2n - 1$. Denote by $l(A)$ the length of A with respect to the set of all reflections, i.e., $l(A)$ is the smallest integer m such that A is a product of m reflections. A reflection is a matrix conjugate to ${\text {diag}}( - 1,1, \ldots ,1)$. Our main result is the formula $l(A) = \max (k(A),k({A^\ast }))$.