  We observe the build up of strong (~50%) spontaneous vector polarisation in
emission from a GaN-based polariton laser excited by short optical pulses at
room temperature. The Stokes vector of emitted light changes its orientation
randomly from one excitation pulse to another, so that the time-integrated
polarisation remains zero. This behaviour is completely different to any
previous laser. We interpret this observation in terms of the spontaneous
symmetry breaking in a Bose-Einstein condensate of exciton-polaritons.
