  A general theorem due to Howe of dual action of a classical group and a
certain non-associative algebra on a space of symmetric or alternating tensors
is reformulated in a setting of second quantization, and familiar examples in
atomic and nuclear physics are discussed. The special case of
orthogonal-orthogonal duality is treated in detail. It is shown that, like it
was done by Helmers more than half a century ago in the analogous case of
symplectic-symplectic duality, one can base a proof of the
orthogonal-orthogonal duality theorem and a precise characterization of the
relation between the equivalence classes of the dually related irreducible
representations on a calculation of characters by combining it with an analysis
of the representation of a reflection. Young diagrams for the description of
equivalence classes of irreducible representations of orthogonal Lie algebras
are introduced. The properties of a reflection of the number non-conserving
part in the dual relationship between orthogonal Lie algebras corroborate a
picture of an almost perfect symmetry between the partners.
