# A $$q$$-weighted Robinson-Schensted algorithm Published on 2013-06-01 | Comments

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:

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.