  This note presents some equalities in law for $Z_N:=\det(\Id-G)$, where $G$
is an element of a subgroup of the set of unitary matrices of size $N$, endowed
with its unique probability Haar measure. Indeed, under some general
conditions, $Z_N$ can be decomposed as a product of independent random
variables, whose laws are explicitly known. Our results can be obtained in two
ways : either by a recursive decomposition of the Haar measure or by previous
results by Killip and Nenciu on orthogonal polynomials with respect to some
measure on the unit circle. This latter method leads naturally to a study of
determinants of a class of principal submatrices.
