Shuffle algebras, lattice paths and Macdonald functions

We consider partition functions on the N×N square lattice with the local Boltzmann weights given by the R-matrix of the Ut(slˆn+1|m) quantum algebra. We identify boundary states such that the square lattice can be viewed on a conic surface. The partition function ZN on this lattice computes the weighted sum over all possible closed coloured lattice paths with n+m different colours: n “bosonic” colours and m “fermionic” colours. Each bosonic (fermionic) path of colour i contributes a factor of zi (wi) to the weight of the configuration. We show the following:i)ZN is a symmetric function in the spectral parameters x1…xN and generates basis elements of the commutative trigonometric Feigin–Odesskii shuffle algebra. The generating function of ZN admits a shuffle-exponential formula analogous to the Macdonald Cauchy kernel.ii)ZN is a symmetric function in two alphabets (z1…zn) and (w1…wm). When x1…xN are set to be equal to the box content of a skew Young diagram μ/ν with N boxes the partition function ZN reproduces the skew Macdonald function Pμ/ν[w−z].