From stylic monoid to Catalan monoid

Source abstract

The stylic monoid $\mathrm{Styl}_n$, introduced by Abram and Reutenauer, is the quotient of the plactic monoid by the relations $x^2=x$, and its elements are represented by $N$-tableaux. Volkov showed that the Catalan monoid $\mathrm{Cat}_n$ of order-preserving, order-decreasing self-maps of $\{0,1,\ldots,n\}$ is a quotient of $\mathrm{Styl}_n$. However, the quotient map is defined on generators, and it is not apparent how to see, from an $N$-tableau, the map in $\mathrm{Cat}_n$ it corresponds to. In this paper we give a simple visual way to read off this map, and some of its main properties, from the $N$-tableau. The new ingredient is that we do not insist on drawing an $N$-tableau as a classical Young tableau: we allow the entries of each row to be shifted relative to the row below, as long as each entry stays above a smaller one. We call this a positioning, and prove that the column word read from any positioning is plactically equivalent to the usual column word; so every positioning can be used to compute the quotient map. We work with the tight positioning, in which each entry is pushed as far right as possible, and define the full core of an $N$-tableau: the part of each column that climbs by consecutive values from the bottom row. We call an $N$-tableau full if it equals its full core. We prove that passing to the full core does not change the image in $\mathrm{Cat}_n$, that full $N$-tableaux are in bijection with $\mathrm{Cat}_n$, and we show how to read the corresponding map directly off a full $N$-tableau.