Transfer matrices for the totally asymmetric simple exclusion process

We consider the totally asymmetric simple exclusion process on a finite lattice with open boundaries. We show, using the recursive structure of the Markov matrix that encodes the dynamics, that there exist two transfer matrices T_{L - 1, L} and that intertwine the Markov matrices of consecutive system sizes: . This semi-conjugation property of the dynamics provides an algebraic counterpart for the matrix-product representation of the steady state of the process.