# A \(q\)-Robinson-Schensted-Knuth algorithm and a \(q\)-polymer

Published on 2016-10-13

(Latest update: 2017-01-12) In Matveev-Petrov 2016 a \(q\)-deformed Robinson-Schensted-Knuth algorithm (\(q\)RSK) was introduced. In this article we give reformulations of this algorithm in terms of Noumi-Yamada description, growth diagrams and local moves. We show that the algorithm is symmetric, namely the output tableaux pair are swapped in a sense of distribution when the input matrix is transposed. We also formulate a \(q\)-polymer model based on the \(q\)RSK and 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 \(q\)RSK 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 (\(q\)PNG) and write down the joint distribution of the \(q\)-polymer partition functions at a fixed time.

This article is available at arXiv. It seems to me that one difference between arXiv and Github is that on arXiv each preprint has a few versions only. In Github many projects have a "dev" branch hosting continuous updates, whereas the master branch is where the stable releases live.

Here is a "dev" version of the article, which I shall push to arXiv when it stablises. Below is the changelog.

- 2017-01-12: Typos and grammar, arXiv v2.
- 2016-12-20: Added remarks on the geometric \(q\)-pushTASEP. Added remarks on the converse of the Burke property. Added natural language description of the \(q\)RSK. Fixed typos.
- 2016-11-13: Fixed some typos in the proof of Theorem 3.
- 2016-11-07: Fixed some typos. The \(q\)-Burke property is now stated in a more symmetric way, so is the law of large numbers Theorem 2.
- 2016-10-20: Fixed a few typos. Updated some references. Added a reference: a set of notes titled "RSK via local transformations". It is written by Sam Hopkins in 2014 as an expository article based on MIT combinatorics preseminar presentations of Alex Postnikov. It contains some idea (applying local moves to a general Young-diagram shaped array in the order that matches any growth sequence of the underlying Young diagram) which I thought I was the first one to write down.