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Description

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.