Role of Sp(12,R) in the harmonic-oscillator quark model

We consider the spectrum-generating algebra Sp(12,$R$) of the harmonic-oscillator quark model and show how it may be used to label the oscillator eigenstates. We give a new and direct method of constructing wave functions of definite symmetry type. We show that Sp(12,$R$) provides the most appropriate means of classifying the symmetry breaking produced by an anharmonic perturbation and we derive an algebraic mass formula for the $N=2\mathrm{}$ multiplets of the model, and indicate how the method may be generalized to arbitrary $N$. Finally we extend the present successful phenomenological analysis of the baryon spectrum performed by Isgur and Karl, and discuss the possibility that the $\Delta \mathrm{D}35\left(1925\right)\mathrm{}$ resonance is evidence for a [56,$1^{-}$] multiplet corresponding to excitation of new gluonic degrees of freedom within baryons.