  Critical exponents of the Kawasaki dynamics in the Ising chain are
re-examined numerically through the spectrum gap of evolution operators
constructed both in spin and domain wall representations. At low temperature
regimes the latter provides a rapid finite-size convergence to these exponents,
which tend to $z \simeq 3.11$ for instant quenches under ferromagnetic
couplings, while approaching to $z \simeq 2$ in the antiferro case. The spin
representation complements the evaluation of dynamic exponents at higher
temperature scales, where the kinetics still remains slow.
