We have studied the ordering of the dynamic q-colours Potts model on a two dimensional square lattice. On the basis of our observations we propose that, if q is large enough, the zero-temperature dynamics is not able to break global and local null-magnetisation symmetries: when q < 4 the system forms domains with a size proportional to the system size while, for q > 4, and provided that the thermodynamic limit is taken before the in nite-time limit, it relaxes towards a non-equilibrium phase with energy larger than the ground state energy, in agreement with the previous ndings of De Oliveira et. al. [2]. On the other hand, at low but positive temperature, the system equilibrates by locally nucleating domains that will eventually grow up to the size of the system, and the nucleation time does not depend on the system size for a big enough system. We characterize quantitatively this process with the help of an order parameter accounting for the local ordering. Finally, we discuss some order parameters for q-degenerated systems and we propose an order parameter constructed according to the representations of the symmetric group. The q > 4 limit for the existence of the glassy phase and its generalisation, q > d + 2, in d dimensions are presented as a consequence of the symmetries of the proposed ordered parameter, and this generalization coincides with the one proposed by Lifshitz [1].

[1] Lifshitz I. M., Zh. Eksp. Teor. Fiz. 42 (1962) (Sov. Phys. JEPT 15 (1962) 939)

[2] M. J. de Oliveira, A. Petri, T. Tom e, Europhys. Lett. 65 20 (2004). M. J. de Oliveira, A. Petri and T. Tom, Physica A 342 (1-2) 97-103 (2004). A. Petri, Braz. J. Phys 33 521 (2003). M. J. de Oliveira, A. Petri, T. Tom e, cond-mat/0402310 (2004).