Relations between reflections in the unitary group and Hermitian hyperquadrics in the complex projective space

AbstractWe consider the subgroup G(n) of the compact unitary group U(n) generated by reflections. By a reflection we mean an element R of U(n) such that R2=I and −1 is a simple eigenvalue of R. It is easy to describe the relations between reflections of the form R1R2=R3R4. One of our main results is that these relations together with R2=I are the defining relations of G(n). Other results characterize the set of all shortest sequences R1, R2,…, Rm of reflections whose product is a fixed element of G(n)