The multispecies TASEP on a circle is a particle model whose partition function is connected to Macdonald polynomials at t=0. Multiline queues were introduced by Ferrari and Martin (2007) to compute the stationary distribution of the TASEP on a circle. Their construction admits a natural interpretation in terms of tensor products of Kirillov–Reshetikhin crystals of affine type A, and a map from these tensor products to TASEP states is given by crystal-theoretic operations. We show that there is a natural Markov chain on the crystal that projects to the TASEP via this map.
The multispecies open boundary TASEP can be viewed as a type C analogue, with its partition function connected to Koornwinder polynomials at t=0. However, computing its stationary distribution beyond rank 1 has remained a longstanding open problem. Using Kirillov–Reshetikhin crystals of affine type C, we construct type C multiline queues and a corresponding Ferrari–Martin pairing algorithm, and show the crystal likewise admits a Markov chain that projects to the TASEP. Our construction is a combinatorial interpretation of the virtualization map for Kashiwara–Nakashima columns. We thus obtain a crystal-theoretic formula for the stationary distribution of the multispecies open boundary TASEP at the specialization alpha=beta=1. This is joint work with Jeronimo Valencia Porras.