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# Copyright (C) 2013-2021 Yuchen Pei.
# Permission is granted to copy, distribute and/or modify this
# document under the terms of the GNU Free Documentation License,
# Version 1.3 or any later version published by the Free Software
# Foundation; with no Invariant Sections, no Front-Cover Texts, and
# no Back-Cover Texts. You should have received a copy of the GNU
# Free Documentation License. If not, see <https://www.gnu.org/licenses/>.
# This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.
#+title: A \(q\)-Robinson-Schensted-Knuth algorithm and a \(q\)-polymer
#+date: <2016-10-13>
(Latest update: 2017-01-12) In
[[http://arxiv.org/abs/1504.00666][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
[[https://arxiv.org/abs/1610.03692][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.
[[file:/assets/qrsklatest.pdf][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: [[http://web.mit.edu/~shopkins/docs/rsk.pdf][a set of notes
titled "RSK via local transformations"]]. It is written by
[[http://web.mit.edu/~shopkins/][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.
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