  We consider a novel approach to the nuclear shell model. The one-dimensional
harmonic oscillator in a box is used to introduce the concept of an
oblique-basis shell-model theory. By implementing the Lanczos method for
diagonalization of large matrices, and the Cholesky algorithm for solving
generalized eigenvalue problems, the method is applied to nuclei. The
mixed-symmetry basis combines traditional spherical shell-model states with
SU(3) collective configurations. We test the validity of this mixed-symmetry
scheme on 24Mg and 44Ti. Results for 24Mg, obtained using the Wilthental USD
intersection in a space that spans less than 10% of the full-space, reproduce
the binding energy within 2% as well as an accurate reproduction of the
low-energy spectrum and the structure of the states - 90% overlap with the
exact eigenstates. In contrast, for an m-scheme calculation, one needs about
60% of the full space to obtain compatible results. Calculations for 44Ti
support the mixed-mode scheme although the pure SU(3) calculations with few
irreps are not as good as the standard m-scheme calculations. The strong
breaking of the SU(3) symmetry results in relatively small enhancements within
the combined basis. However, an oblique-basis calculation in 50% of the full
pf-shell space is as good as a usual m-scheme calculation in 80% of the space.
Results for the lower pf-shell nuclei 44-48Ti and 48Cr, using the Kuo-Brown-3
interaction, show that SU(3) symmetry breaking in this region is driven by the
single-particle spin-orbit splitting. In our study we observe some interesting
coherent structures, such as coherent mixing of basis states,
quasi-perturbative behavior in the toy model, and enhanced B(E2) strengths
close to the SU(3) limit even though SU(3) appears to be rather badly broken.
