Exact Results for Spin and Charge Dynamics of Electrons with Supersymmetry

Excitation spectra in the SU($\nu +1$,1) supersymmetric t-J model with long-range exchange and transfer has quadratic dependence on spin and charge currents for all energies. After brief review on the supersymmetry, this paper gives a simple explanation for the spin-charge separation on the basis of a first-quantized representation. A useful identity is derived on permutation properties of SU($\nu$) Jastrow-type wave functions, which constitute the ground states for a class of 1/$r^2$ interaction models.