# An algebraic construction of duality functions for the stochastic
U_q(A_n^{(1)}) vertex model and its degenerations

Research paper by **Jeffrey Kuan**

Indexed on: **16 Jan '17**Published on: **16 Jan '17**Published in: **arXiv - Mathematics - Probability**

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#### Abstract

A recent paper \cite{KMMO} introduced the stochastic U_q(A_n^{(1)}) vertex
model. The stochastic S-matrix is related to the R-matrix of the quantum group
U_q(A_n^{(1)}) by a gauge transformation. We will show that a certain function
D^+_{\mu} intertwines with the transfer matrix and its space reversal. When
interpreting the transfer matrix as the transition matrix of a discrete-time
totally asymmetric particle system on the one-dimensional lattice Z, the
function D^+_{\mu} becomes a Markov duality function D_{\mu} which only depends
on q and the vertical spin parameters \mu_x. By considering degenerations in
the spectral parameter, the duality results also hold on a finite lattice with
closed boundary conditions, and for a continuous-time degeneration. This
duality function had previously appeared in a multi-species ASEP(q,j) process.
The proof here uses that the R-matrix intertwines with the co-product, but does
not explicitly use the Yang-Baxter equation.
It will also be shown that the stochastic U_q(A_n^{(1)}) is a multi-species
version of a stochastic vertex model studied in \cite{BP,CP}. This will be done
by generalizing the fusion process of \cite{CP} and showing that it matches the
fusion of \cite{KRL} up to the gauge transformation.
We also show, by direct computation, that the multi-species q-Hahn Boson
process (which arises at a special value of the spectral parameter) also
satisfies duality with respect to D_0, generalizing the single-species result
of \cite{C}.