  We study a model of dilute oriented loops on the square lattice, where each
loop is compatible with a fixed, alternating orientation of the lattice edges.
This implies that loop strands are not allowed to go straight at vertices, and
results in an enhancement of the usual O(n) symmetry to U(n). The corresponding
transfer matrix acts on a number of representations (standard modules) that
grows exponentially with the system size. We derive their dimension and those
of the centraliser by both combinatorial and algebraic techniques. A mapping
onto a field theory permits us to identify the conformal field theory governing
the critical range, $n \le 1$. We establish the phase diagram and the critical
exponents of low-energy excitations. For generic n, there is a critical line in
the universality class of the dilute O(2n) model, terminating in an SU(n+1)
point. The case n=1 maps onto the critical line of the six-vertex model, along
which exponents vary continuously.
