A $q$-Robinson-Schensted-Knuth Algorithm and a $q$-Polymer

In Matveev-Petrov (2017) a q-deformed Robinson-Schensted-Knuth algorithm (qRSK) was introduced. In this article we give reformulations of this algorithm in terms of the Noumi-Yamada description, growth diagrams and local moves. We show that the algorithm is symmetric, namely the output tableaux pairs are swapped in a sense of distribution when the input matrix is transposed. We also formulate a q-polymer model based on the qRSK, prove the corresponding Burke property, which we use to show a strong law of large numbers for the partition function given stationary boundary conditions and q-geometric weights. We use the q-local moves to define a generalisation of the qRSK taking a Young diagram-shape of array as the input. We write down the joint distribution of partition functions in the space-like direction of the q-polymer in q-geometric environment, formulate a q-version of the multilayer polynuclear growth model (qPNG) and write down the joint distribution of the q-polymer partition functions at a fixed time.