OVERLAPS BETWEEN THE IRREDUCIBLE REPRESENTATIONS OF 2 SO(7) SUBGROUPSOF SO(8) USED IN THE QUARK-MODEL OF THE ATOMIC-F SHELL

In his studies of f electrons in atoms, Racah introduced the group SO( 7) and its subgroup G2, with irreducible representations (irreps) W an d U. By using a quarklike basis, these groups can be conveniently embe dded in SO(8). This larger group, with irreps V, possesses two other S O(7) groups as subgroups that themselves contain G2 as a common subgro up. One of them, SO(7)' (with irreps W'), has been used to derive new selection rules on operators of physical interest. We describe methods for calculating the overlaps (VWU\VW'U), the ultimate aim being to fa cilitate the transformations between SO(7) and SO(7)'. A table of rele vant 6-U symbols (the G2 generalizations of 6-j symbols) is given. Whe n V possesses null triality (that is, when the symbols labelling the o pen ends of the Dynkin diagram for SO(8) are equal), an undetermined p hase in the overlaps can be used to generate matrix representations of S3, the permutation group on three objects. A brief table of zero ove rlaps is given. A remarkable factorization of the overlaps ((4310)W(40 )\(4310)W'(40)) is noted, where (4310) is the irrep of SO(8) with dime nsion 25725.