The Sorting Order on a Coxeter Group
Copyright policy )The Sorting Order on a Coxeter Group
Let $(W,S)$ be an arbitrary Coxeter system. For each sequence $\omega =(\omega_1,\omega_2,\ldots) \in S^{\ast}$ in the generators we define a partial order― called the $\omega \mathsf{-sorting order}$ ―on the set of group elements $W_{\omega} \subseteq W$ that occur as finite 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 strong Bruhat orders on the group. Moreover, the $\omega$-sorting order is a "maximal lattice'' in the sense that the addition of any collection of edges from the Bruhat order results in a nonlattice. Along the way we define a class of structures called $\mathsf{supersolvable}$ $\mathsf{antimatroids}$ and we show that these are equivalent to the class of supersolvable join-distributive lattices.
- University of Minnesota United States
- University of Minesota United States
- University of Minnesota-Twin Cities United States
- University of Minnesota System United States
- School of Mathematics University of Minnesota United States
partial orders, Group Theory (math.GR), Bruhat order, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], Abstract convex geometry, Catalan number, Reflection and Coxeter groups (group-theoretic aspects), [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Other generalizations of distributive lattices, Discrete Mathematics and Combinatorics, supersolvable lattices, Sorting algorithm, Coxeter group, coxeter group, Combinatorial aspects of groups and algebras, antimatroids, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], Computational Theory and Mathematics, Partial order, 20F55, Combinatorics (math.CO), Mathematics - Group Theory, join-distributive lattices, antimatroid, convex geometry, supersolvable lattice, Supersolvable lattice, join-distributive lattice, [info.info-dm] computer science [cs]/discrete mathematics [cs.dm], Coxeter groups, sorting algorithms, Theoretical Computer Science, Combinatorics of partially ordered sets, 20F55; 06A07, Antimatroid, QA1-939, FOS: Mathematics, Mathematics - Combinatorics, Join-distributive lattice, [math.math-co] mathematics [math]/combinatorics [math.co], Lattice, 06A07, weak order, Catalan numbers, Mathematics
partial orders, Group Theory (math.GR), Bruhat order, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], Abstract convex geometry, Catalan number, Reflection and Coxeter groups (group-theoretic aspects), [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Other generalizations of distributive lattices, Discrete Mathematics and Combinatorics, supersolvable lattices, Sorting algorithm, Coxeter group, coxeter group, Combinatorial aspects of groups and algebras, antimatroids, [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], Computational Theory and Mathematics, Partial order, 20F55, Combinatorics (math.CO), Mathematics - Group Theory, join-distributive lattices, antimatroid, convex geometry, supersolvable lattice, Supersolvable lattice, join-distributive lattice, [info.info-dm] computer science [cs]/discrete mathematics [cs.dm], Coxeter groups, sorting algorithms, Theoretical Computer Science, Combinatorics of partially ordered sets, 20F55; 06A07, Antimatroid, QA1-939, FOS: Mathematics, Mathematics - Combinatorics, Join-distributive lattice, [math.math-co] mathematics [math]/combinatorics [math.co], Lattice, 06A07, weak order, Catalan numbers, Mathematics
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