  We consider the model proposed by Axelrod for dissemination of cultures on a
2-dimensional squared lattice. We review this model from an analytic point of
view. We define $\left\langle s(t)\right\rangle$ to quantify possible culture
configurations at time $t$ in a society. Typical initial culture configurations
of this model are characterised. Equation of motion in terms of $\left\langle
s(t)\right\rangle$ is derived. We study the graph of development of this
Axelrod system toward to its culture configurations equilibrium. Generically,
we observe that this model undergoes three phases of development. We give a
quantitative explanation about these three different phases of development.
  Keeping up with this Axelrod model, we characterize its culture
configurations space at equilibrium point where $\left\langle
s(t_{\text{eq}})\right\rangle = 1$. This space is called monoculture space.
Understanding this space is equivalent to restrict to the space of culture
configurations from one individual in the model. This individual culture space
is identified to the space $V_N^{\otimes F}$ up to isomorphisms. Action of the
permutation group $S_N$ on the space $V_{N}^{\otimes F}$ is considered. Under
this action, the observable $\left\langle s(t)\right\rangle$ is an invariant of
the Axelrod system. We explore this symmetry and classify the different
inequivalent classes of culture configurations composing the monoculture space.
To achieve this, we consider the case $N\geq F$. We propose techniques from
group representation theory to perform this classification. The inequivalent
classes of culture configurations are indexed by the Dickau diagrams which are
associated to the Bell number $B_F$. A concrete example with $F=4$ and $N\geq
4$ is considered for a full illustration of our analysis.
