Shuffle algebras, lattice paths and Macdonald functions

Abstract

We consider partition functions on the $N\times N$ square lattice with the local Boltzmann weights given by the R-matrix of the $U_t\left({\widehat{sl}}_{n+1|m}\right)$ quantum algebra. We identify boundary states such that the square lattice can be viewed on a conic surface. The partition function $Z_N$ 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 $z_i$ ($w_i$) to the weight of the configuration. We show the following:

• i)
  $Z_N$ is a symmetric function in the spectral parameters $x_1\dots x_N$ and generates basis elements of the commutative trigonometric Feigin–Odesskii shuffle algebra. The generating function of $Z_N$ admits a shuffle-exponential formula analogous to the Macdonald Cauchy kernel.
• ii)
  $Z_N$ is a symmetric function in two alphabets $(z_1\dots z_n)$ and $(w_1\dots w_m)$. When $x_1\dots x_N$ are set to be equal to the box content of a skew Young diagram $\mu /\nu$ with N boxes the partition function $Z_N$ reproduces the skew Macdonald function $P_{\mu /\nu }[w-z]$.