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arXiv · 0712.1047

The sorting order on a Coxeter group

Drew Armstrong
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Abstract

Let $(W,S)$ be an arbitrary Coxeter system. For each word $ω$ in the generators we define a partial order--called the {\sf $ω$-sorting order}--on the set of group elements $W_ω\subseteq W$ that occur as subwords of $ω$. We show that the $ω$-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and Bruhat orders on the group. Moreover, the $ω$-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.

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Drew Armstrong. 2009-03-30. The sorting order on a Coxeter group. https://arxiv.org/abs/0712.1047

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