Abstract
Let
(
W
,
S
)
be an arbitrary Coxeter system. For each word
ω in the generators we define a partial order—called the
ω-
sorting order—on the set of group elements
W
ω
⊆
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
supersolvable antimatroids and we show that these are equivalent to the class of supersolvable join-distributive lattices.
