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 W ω ⊆ 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 language | English (US) |
|---|---|
| Pages (from-to) | 1285-1305 |
| Number of pages | 21 |
| Journal | Journal of Combinatorial Theory, Series A |
| Volume | 116 |
| Issue number | 8 |
| DOIs | |
| State | Published - Nov 2009 |
| Externally published | Yes |
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Keywords
- Abstract convex geometry
- Antimatroid
- Catalan number
- Coxeter group
- Join-distributive lattice
- Lattice
- Partial order
- Sorting algorithm
- Supersolvable lattice
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Theoretical Computer Science
- Computational Theory and Mathematics
Cite this
The sorting order on a Coxeter group. / Armstrong, Drew.
In: Journal of Combinatorial Theory, Series A, Vol. 116, No. 8, 11.2009, p. 1285-1305.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - The sorting order on a Coxeter group
AU - Armstrong, Drew
PY - 2009/11
Y1 - 2009/11
N2 - 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 W ω ⊆ 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.
AB - 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 W ω ⊆ 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.
KW - Abstract convex geometry
KW - Antimatroid
KW - Catalan number
KW - Coxeter group
KW - Join-distributive lattice
KW - Lattice
KW - Partial order
KW - Sorting algorithm
KW - Supersolvable lattice
UR - http://www.scopus.com/inward/record.url?scp=70349309916&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=70349309916&partnerID=8YFLogxK
U2 - 10.1016/j.jcta.2009.03.009
DO - 10.1016/j.jcta.2009.03.009
M3 - Article
VL - 116
SP - 1285
EP - 1305
JO - Journal of Combinatorial Theory - Series A
JF - Journal of Combinatorial Theory - Series A
SN - 0097-3165
IS - 8
ER -