diff options
Diffstat (limited to 'posts/2013-06-01-q-robinson-schensted-paper.md')
| -rw-r--r-- | posts/2013-06-01-q-robinson-schensted-paper.md | 31 |
1 files changed, 31 insertions, 0 deletions
diff --git a/posts/2013-06-01-q-robinson-schensted-paper.md b/posts/2013-06-01-q-robinson-schensted-paper.md new file mode 100644 index 0000000..657412d --- /dev/null +++ b/posts/2013-06-01-q-robinson-schensted-paper.md @@ -0,0 +1,31 @@ +--- +template: oldpost +title: A \(q\)-weighted Robinson-Schensted algorithm +date: 2013-06-01 +comments: true +tags: RS, \(q\)-Whittaker_functions, Macdonald_polynomials +archive: false +--- +In [this paper](https://projecteuclid.org/euclid.ejp/1465064320) with [Neil](http://www.bristol.ac.uk/maths/people/neil-m-oconnell/) we construct a \\(q\\)-version of the Robinson-Schensted +algorithm with column insertion. Like the [usual RS +correspondence](http://en.wikipedia.org/wiki/Robinson–Schensted_correspondence) +with column insertion, this algorithm could take words as input. Unlike +the usual RS algorithm, the output is a set of weighted pairs of +semistandard and standard Young tableaux \\((P,Q)\\) with the same +shape. The weights are rational functions of indeterminant \\(q\\). + +If \\(q\\in\[0,1\]\\), the algorithm can be considered as a randomised +RS algorithm, with 0 and 1 being two interesting cases. When +\\(q\\to0\\), it is reduced to the latter usual RS algorithm; while +when \\(q\\to1\\) with proper scaling it should scale to directed random +polymer model in [(O'Connell 2012)](http://arxiv.org/abs/0910.0069). +When the input word \\(w\\) is a random walk: + +\\begin{align\*}\\mathbb +P(w=v)=\\prod\_{i=1}^na\_{v\_i},\\qquad\\sum\_ja\_j=1\\end{align\*} + +the shape of output evolves as a Markov chain with kernel related to +\\(q\\)-Whittaker functions, which are Macdonald functions when +\\(t=0\\) with a factor. + + |