From 147a19e84a743f1379f05bf2f444143b4afd7bd6 Mon Sep 17 00:00:00 2001 From: Yuchen Pei Date: Fri, 18 Jun 2021 12:58:44 +1000 Subject: Updated. --- .../2013-06-01-q-robinson-schensted-paper.html | 52 ++++++++++++++++++++++ 1 file changed, 52 insertions(+) create mode 100644 site-from-md/posts/2013-06-01-q-robinson-schensted-paper.html (limited to 'site-from-md/posts/2013-06-01-q-robinson-schensted-paper.html') diff --git a/site-from-md/posts/2013-06-01-q-robinson-schensted-paper.html b/site-from-md/posts/2013-06-01-q-robinson-schensted-paper.html new file mode 100644 index 0000000..2ccaf87 --- /dev/null +++ b/site-from-md/posts/2013-06-01-q-robinson-schensted-paper.html @@ -0,0 +1,52 @@ + + + + + A \(q\)-weighted Robinson-Schensted algorithm + + + + + +
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A \(q\)-weighted Robinson-Schensted algorithm

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Posted on 2013-06-01

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In this paper with Neil we construct a \(q\)-version of the Robinson-Schensted algorithm with column insertion. Like the usual RS 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 \(q\).

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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). When the input word \(w\) is a random walk:

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\begin{align*}\mathbb P(w=v)=\prod_{i=1}^na_{v_i},\qquad\sum_ja_j=1\end{align*}

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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.

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