On the use of the group SO(4,2) in atomic and molecular physics

In this paper the dynamical noninvariance group SO(4,2) for a hydrogen-like atom is derived through two different approaches. The first one is by an established traditional ascent process starting from the symmetry group SO(3). This approach is presented in a mathematically oriented original way with a special emphasis on maximally superintegrable systems, N-dimensional extension and little groups. The second approach is by a new symmetry descent process starting from the noninvariance dynamical group Sp(8,R) for a four-dimensional harmonic oscillator. It is based on the little known concept of a Lie algebra under constraints and corresponds in some sense to a symmetry breaking mechanism. This paper ends with a brief discussion of the interest of SO(4,2) for a new group-theoretical approach to the periodic table of chemical elements. In this connection, a general ongoing programme based on the use of a complete set of commuting operators is briefly described. It is believed that the present paper could be useful not only to the atomic and molecular community but also to people working in theoretical and mathematical physics.Comment: 31 page

REDUCED NOTATION, INNER PLETHYSMS AND THE SYMMETRICAL GROUP

International audienceThe reduced notation for irreducible representations of the symmetric group S-n is interpreted in terms of symmetric formal series and vertex operators, and is used to prove a number of properties of reduced Kronecker products and inner plethysms in an n-independent manner. Conditions for self-associativity of Kronecker products and inner plethysms are established. Reduced inner plethysms are developed and applied to the question of non-simple phase groups among the symmetric S-n and alternating A(n) groups