  We consider a multi-species generalization of the totally asymmetric simple
exclusion process (TASEP) with the simple hopping rule: for x and yth-class
particles (x<y), the transition xy -> yx occurs with a rate independent from
the values x and y. P. A. Ferrari and J. Martin (2007) obtained the stationary
state of this model thanks to a combinatorial algorithm, which was subsequently
interpreted as a matrix product representation by Evans et al. (2009). This
`matrix ansatz' shows that the stationary state of the multi-species TASEP with
N classes of particles (N-TASEP) can be constructed algebraically by the action
of an operator on the (N-1)-TASEP stationary state. Besides, Arita et al.
(2009) analyzed the spectral structure of the Markov matrix: they showed that
the set of eigenvalues of the N-TASEP contains those of the (N-1)-TASEP and
that the various spectral inclusions can be encoded in a hierarchical
set-theoretic structure known as the Hasse diagram. Inspired by these works, we
define nontrivial operators that allow us to construct eigenvectors of the
N-TASEP by lifting the eigenvectors of the (N-1)-TASEP. This goal is achieved
by generalizing the matrix product representation and the Ferrari-Martin
algorithm. In particular, we show that the matrix ansatz is not only a
convenient tool to write the stationary state but in fact intertwines Markov
matrices of different values of N.
