  Real or complex tensor model observables, the backbone of the tensor theory
space, are classical (unitary, orthogonal, symplectic) Lie group invariants.
These observables represent as colored graphs, and that representation gives an
handle to study their combinatorial, topological and algebraic properties. We
give here an overview of the symmetric group-theoretic formulation of the
enumeration of unitary and orthogonal invariant observables which turns out to
bear a rich structure. From their counting formulae, one finds a correspondence
with topological field theory on 2-cellular complexes that brings other
interpretations of the same countings. Furthermore, tensor model observables
span an algebra that turns out to be semi-simple. Dealing with complex tensors,
we discuss the representation theoretic base of the algebra making explicit its
Wedderburn-Artin decomposition. The real case is more subtle as a base of its
Wedderburn-Artin decomposition is yet unknown.
