The sorting order on a Coxeter group

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19 Scopus citations


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 Wω ⊆ 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.

Original languageEnglish (US)
Pages (from-to)1285-1305
Number of pages21
JournalJournal of Combinatorial Theory. Series A
Issue number8
StatePublished - Nov 2009
Externally publishedYes


  • Abstract convex geometry
  • Antimatroid
  • Catalan number
  • Coxeter group
  • Join-distributive lattice
  • Lattice
  • Partial order
  • Sorting algorithm
  • Supersolvable lattice

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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