  Svetitsky and Yaffe have argued that -- if the deconfinement phase transition
of a (d+1)-dimensional Yang-Mills theory with gauge group G is second order --
it should be in the universality class of a d-dimensional scalar model
symmetric under the center C(G) of G. These arguments have been investigated
numerically only considering Yang-Mills theory with gauge symmetry in the
G=SU(N) branch, where C(G)=Z(N). The symplectic groups Sp(N) provide another
extension of SU(2)=Sp(1) to general N and they all have the same center Z(2).
Hence, in contrast to the SU(N) case, Sp(N) Yang-Mills theory allows to study
the relevance of the group size on the order of the deconfinement phase
transition keeping the available universality class fixed. Using lattice
simulations, we present numerical results for the deconfinement phase
transition in Sp(2) and Sp(3) Yang-Mills theories both in (2+1)d and (3+1)d. We
then make a conjecture on the order of the deconfinement phase transition in
Yang-Mills theories with general Lie groups SU(N), SO(N), Sp(N) and with
exceptional groups G(2), F(4), E(6), E(7), E(8). Numerical results for G(2)
Yang-Mills theory at finite temperature in (3+1)d are also presented.
