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#+title: A \(q\)-weighted Robinson-Schensted algorithm

#+date: <2013-06-01>

In [[https://projecteuclid.org/euclid.ejp/1465064320][this paper]] with
[[http://www.bristol.ac.uk/maths/people/neil-m-oconnell/][Neil]] we
construct a \(q\)-version of the Robinson-Schensted algorithm with
column insertion. Like the
[[http://en.wikipedia.org/wiki/Robinson–Schensted_correspondence][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
\(q\).

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
[[http://arxiv.org/abs/0910.0069][(O'Connell 2012)]]. 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.