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We introduce an overcomplete highest weight state basis as a calculationaltool for the description of the isotropic spin-1 chain with bilinear exchangecoupling J1 and biquadratic coupling J2. The ground state can be expressedexactly at the three special points in the phase diagram where the Hamiltoniancorresponds to a sum of nearest neighbor total spin projection operators(J1=0>J2, J1=-J2<0, and J1=-J2/3<0). In particular, at the phase transitionpoint J1=-J2<0 it is possible to exactly compute the ground states, excitedstates, expectation values, and correlation functions by using the new totalspin basis.

Highest weight state description of the isotropic spin-1 chain