 Main
Stacksorting for Coxeter groups
 Author(s): Defant, Colin
 et al.
Abstract
Given an essential semilattice congruence $\equiv$ on the left weak order of a \linebreak Coxeter group $W$, we define the Coxeter stacksorting operator ${\bf S}_\equiv:W\to W$ by ${{\bf S}_\equiv(w)=w\big(\pi_\downarrow^\equiv(w)\big)^{1}}$, where $\pi_\downarrow^\equiv(w)$ is the unique minimal element of the congruence class of $\equiv$ containing $w$. When $\equiv$ is the sylvester congruence on the symmetric group $S_n$, the operator ${\bf S}_\equiv$ is West's stacksorting map. When $\equiv$ is the descent congruence on $S_n$, the operator ${\bf S}_\equiv$ is the popstacksorting map. We establish several general results about Coxeter stacksorting operators, especially those acting on symmetric groups. For example, we prove that if $\equiv$ is an essential lattice congruence on $S_n$, then every permutation in the image of ${\bf S}_\equiv$ has at most $\left\lfloor\frac{2(n1)}{3}\right\rfloor$ right descents; we also show that this bound is tight. We then introduce analogues of permutree congruences in types $B$ and $\widetilde A$ and use them to isolate Coxeter stacksorting operators $\mathtt{s}_B$ and $\widetilde{\mathtt{s}}$ that serve as canonical type$B$ and type$\widetilde A$ counterparts of West's stacksorting map. We prove analogues of many known results about West's stacksorting map for the new operators $\mathtt{s}_B$ and $\widetilde{\mathtt{s}}$. For example, in type $\widetilde A$, we obtain an analogue of Zeilberger's classical formula for the number of $2$stacksortable permutations in $S_n$.
Mathematics Subject Classifications: 06A12, 06B10, 37E15, 05A05, 05E16
Main Content
Enter the password to open this PDF file:













