A \(q\)-weighted Robinson-Schensted algorithm

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 \(q\).

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:

\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.