TASEP in any Weyl Group

We investigate a Markov chain defined by Thomas Lam, which generalizes the multi-type TASEP on a ring to any Weyl group. For groups of type C we define an analogue of the multiline queues of Ferrari and Martin (which compute the stationary distribution for the classical TASEP). While our construction does not suffice for finding the stationary distribution, the construction gives the stationary distribution of a certain projection of Lam's chain. Also, our approach is incremental, in the sense that the construction appears to fit into a pattern of 'conjugation matrices', which remains to be fully worked out. Finally, we prove a theorem for the classical TASEP which fits into the picture of viewing TASEP in a permutation-free way.