Symmetries and deformations in the spherical shell model

23 pages, 7 figures, 2 tables, to be published in a special edition of Physica Scripta to commemorate the 40e anniversary of the Nobel Prize of Bohr, Mottelson and Rainwater. Some figures may appear in colour only in the online journalpour commémorer le 40me anniversaire du prix Nobel de Bohr, Mottelson et Rainwater.International audienceWe discuss symmetries of the spherical shell model that make contact with the geometric collective model of Bohr and Mottelson. The most celebrated symmetry of this kind is SU(3), which is the basis of Elliott's model of rotation. It corresponds to a deformed mean field induced by a quadrupole interaction in a single major oscillator shell N and can be generalized to include several major shells. As such, Elliott's SU(3) model establishes the link between the spherical shell model and the (quadrupole component of the) geometric collective model. We introduce the analogue symmetry induced by an octupole interaction in two major oscillator shells N-1 and N, leading to an octupole-deformed solution of the spherical shell model. We show that in the limit of large oscillator shells (large N) the algebraic octupole interaction tends to that of the geometric collective model