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<h2> A \(q\)-weighted Robinson-Schensted algorithm </h2>
<p>Posted on 2013-06-01</p>
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<p>In <a href="https://projecteuclid.org/euclid.ejp/1465064320">this paper</a> with <a href="http://www.bristol.ac.uk/maths/people/neil-m-oconnell/">Neil</a> we construct a \(q\)-version of the Robinson-Schensted algorithm with column insertion. Like the <a href="http://en.wikipedia.org/wiki/Robinson–Schensted_correspondence">usual RS correspondence</a> 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 \(q\).</p>
<p>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 <a href="http://arxiv.org/abs/0910.0069">(O’Connell 2012)</a>. When the input word \(w\) is a random walk:</p>
<p>\begin{align*}\mathbb P(w=v)=\prod_{i=1}^na_{v_i},\qquad\sum_ja_j=1\end{align*}</p>
<p>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.</p>
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