Abstract. We prove that the normalisation of the stationary state of the multispecies asymmetric simple exclusion process (mASEP) is a specialisation of a Koornwinder polynomial. As a corollary we obtain that the normalisation of mASEP factorises as a product over multiple copies of the two-species ASEP.Koornwinder polynomials and the stationary open mASEP 2

Abstract. We introduce a new family of symmetric multivariate polynomials, whose coefficients are meromorphic functions of two parameters (q, t) and polynomial in a further two parameters (u, v). We evaluate these polynomials explicitly as a matrix product. At u = v = 0 they reduce to Macdonald polynomials, while at q = 0, u = v = s they recover a family of inhomogeneous symmetric functions originally introduced by Borodin.

We consider the integrable dilute Temperley-Lieb (dTL) O(n = 1) loop model on a semi-infinite strip of finite width L. In the analogy with the Temperley-Lieb (TL) O(n = 1) loop model the ground state eigenvector of the transfer matrix is studied by means of a set of q-difference equations, sometimes called the qKZ equations. We compute some ground state components of the transfer matrix of the dTL model, and show that all ground state components can be recovered for arbitrary L using the qKZ equation and certain recurrence relation. The computations are done for generic open boundary conditions.