The sorting order on a Coxeter group

TL;DR

This paper introduces the $ ext{ extsf{$oldsymbol{ ext{ω}}}$}$-sorting order on Coxeter groups, showing it forms a supersolvable join-distributive lattice that lies between the weak and Bruhat orders, and explores related structures.

Contribution

It defines the $ ext{ extsf{$oldsymbol{ ext{ω}}}$}$-sorting order, proves its lattice properties, and introduces supersolvable antimatroids, establishing new connections in Coxeter group theory.

Findings

$ ext{ extsf{$oldsymbol{ ext{ω}}}$}$-sorting order is a supersolvable join-distributive lattice.

The $ ext{ extsf{$oldsymbol{ ext{ω}}}$}$-sorting order lies strictly between weak and Bruhat orders.

Adding Bruhat covers beyond the $ ext{ extsf{$oldsymbol{ ext{ω}}}$}$-sorting order breaks lattice structure.

Abstract

Let $(W,S)$ be an arbitrary Coxeter system. For each word $\omega$ in the generators we define a partial order--called the {\sf $\omega$-sorting order}--on the set of group elements $W_\omega\subseteq W$ that occur as subwords of $\omega$. We show that the $\omega$-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and Bruhat orders on the group. Moreover, the $\omega$-sorting order is a "maximal lattice" in the sense that the addition of any collection of Bruhat covers results in a nonlattice. Along the way we define a class of structures called {\sf supersolvable antimatroids} and we show that these are equivalent to the class of supersolvable join-distributive lattices.