I don't recall ever seeing a term to refer to Markov chains for which all the transition probabilities matrices are equal ($P^{(n)}=P\quad\forall\,n\in\mathbb{N}$) but I'm sure there should be one.... We statisticians like to put names to stuff.
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$\begingroup$ An über-mixing Markov chain?! Since it converges to the stationary in at most $n$ steps... $\endgroup$– Xi'anDec 3, 2014 at 15:41
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$\begingroup$ Thanks @Xi'an your answer made me realize that my question was not clear. I've edited for clarity. $\endgroup$– PedroDec 4, 2014 at 1:52
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In the book Probability Measures on Semigroups: Convolution Products, Random Walks and Random Matrices, by Göran Högnäs and Arunava Mukherjea (2010), I found the result that, if $P$ is an idempotent stochastic matrix, then it is made of block diagonal matrices with identical rows (Theorem 1.16, p.48). This is also straightforward when considering that $$\lim_n P^n=P,$$ which is then the stationary distribution on the irreducible components of the state space. (And there cannot be a transient component in this case.)
