  In this work we define two kinds of crested product for reversible Markov
chains, which naturally appear as a generalization of the case of crossed and
nested product, as in association schemes theory, even if we do a construction
that seems to be more general and simple. Although the crossed and nested
product are inspired by the study of Gelfand pairs associated with the direct
and the wreath product of two groups, the crested products are a more general
construction, independent from the Gelfand pairs theory, for which a complete
spectral theory is developed. Moreover, the $k$-step transition probability is
given. It is remarkable that these Markov chains describe some classical models
(Ehrenfest diffusion model, Bernoulli--Laplace diffusion model, exclusion
model) and give some generalization of them. As a particular case of nested
product, one gets the classical Insect Markov chain on the ultrametric space.
Finally, in the context of the second crested product, we present a
generalization of this Markov chain to the case of many insects and give the
corresponding spectral decomposition.
