The sorting order on a Coxeter group

  • Authors:

  • Drew Armstrong

  • Affiliations:

  • School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455, United States

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2009

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Abstract

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