The sorting order on a Coxeter group

Details The sorting order on a Coxeter group The sorting order on a Coxeter group

2007-12-07

arxiv.org/abs/0712.1047v2

Mathematics

Combinatorics

34 pages, 7 figures. Final version, to appear in Journal of Combinatorial Theory Series A

Scientific paper

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.

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