We report an ultrasonic investigation of the elastic moduli on a single crystal of hexagonal YMnO_3 as a function of temperature. Stiffening anomalies in the antiferromagnetic Neel state below T_N = 72.4 K are observed on all the four elastic moduli C_{ii}. The anomalies are the most important on C_{11} and C_{66} for in-plane elastic deformations; this is consistent with a strong coupling of the lattice with the in-plane exchange interactions. We use a Landau free energy model to account for these elastic anomalies. We derive an expression which relates the temperature profile of the anomaly to the order parameter; the critical exponent associated to this parameter $beta$ = 0.42 is not consistent with a chiral XY or 3D Heisenberg universality class, but more in agreement with a conventional antiferromagnetic long range order. A tiny softening anomaly on C_{11} for which hysteresis effects are observed could be indicative of an interaction between ferroelectric and magnetic domains at T_N. Moreover, magnetic fluctuations effects both above and below T_N are identified through abnormal temperature and magnetic field effects.
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