The sorting order on a Coxeter group |
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Authors: | Drew Armstrong |
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Institution: | School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, MN 55455, United States |
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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 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. |
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Keywords: | Coxeter group Partial order Lattice Antimatroid Abstract convex geometry Supersolvable lattice Join-distributive lattice Catalan number Sorting algorithm |
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