### Abstract

Let (W,S) be an arbitrary Coxeter system. For each sequence ω = (ω _{1}, ω _{2},. .) ∈ S* 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 finite subwords of !. We show that the ω-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 ω-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 supersolvable antimatroids and we show that these are equivalent to the class of supersolvable join-distributive lattices.

Original language | English (US) |
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Title of host publication | FPSAC'08 - 20th International Conference on Formal Power Series and Algebraic Combinatorics |

Pages | 411-416 |

Number of pages | 6 |

State | Published - 2008 |

Externally published | Yes |

Event | 20th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'08 - Valparaiso, Chile Duration: Jun 23 2008 → Jun 27 2008 |

### Other

Other | 20th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'08 |
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Country | Chile |

City | Valparaiso |

Period | 6/23/08 → 6/27/08 |

### Fingerprint

### Keywords

- Antimatroid
- Convex geometry
- Coxeter group
- Join-distributive lattice
- Supersolvable lattice

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*FPSAC'08 - 20th International Conference on Formal Power Series and Algebraic Combinatorics*(pp. 411-416)

**The sorting order on a Coxeter group.** / Armstrong, Drew.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*FPSAC'08 - 20th International Conference on Formal Power Series and Algebraic Combinatorics.*pp. 411-416, 20th International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'08, Valparaiso, Chile, 6/23/08.

}

TY - GEN

T1 - The sorting order on a Coxeter group

AU - Armstrong, Drew

PY - 2008

Y1 - 2008

N2 - Let (W,S) be an arbitrary Coxeter system. For each sequence ω = (ω 1, ω 2,. .) ∈ S* 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 finite subwords of !. We show that the ω-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 ω-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 supersolvable antimatroids and we show that these are equivalent to the class of supersolvable join-distributive lattices.

AB - Let (W,S) be an arbitrary Coxeter system. For each sequence ω = (ω 1, ω 2,. .) ∈ S* 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 finite subwords of !. We show that the ω-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 ω-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 supersolvable antimatroids and we show that these are equivalent to the class of supersolvable join-distributive lattices.

KW - Antimatroid

KW - Convex geometry

KW - Coxeter group

KW - Join-distributive lattice

KW - Supersolvable lattice

UR - http://www.scopus.com/inward/record.url?scp=84860442284&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860442284&partnerID=8YFLogxK

M3 - Conference contribution

SP - 411

EP - 416

BT - FPSAC'08 - 20th International Conference on Formal Power Series and Algebraic Combinatorics

ER -