Abstract
In this note we construct a poset map from a Boolean algebra to the Bruhat order which unveils an interesting connection between subword complexes, sorting orders, and certain totally nonnegative spaces. This relationship gives a simple new proof that the proper part of Bruhat order is homotopy equivalent to the proper part of a Boolean algebra - that is, to a sphere. We also obtain a geometric interpretation for sorting orders. We conclude with two new results: that the intersection of all sorting orders is the weak order, and the union of sorting orders is the Bruhat order.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 46-53 |
| Number of pages | 8 |
| Journal | Advances in Applied Mathematics |
| Volume | 46 |
| Issue number | 1-4 |
| DOIs | |
| State | Published - Jan 2011 |
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Keywords
- Bruhat order
- Homotopy type
- Quillen Fiber Lemma
- Weak order
ASJC Scopus subject areas
- Applied Mathematics
Cite this
Sorting orders, subword complexes, Bruhat order and total positivity. / Armstrong, Drew; Hersh, Patricia.
In: Advances in Applied Mathematics, Vol. 46, No. 1-4, 01.2011, p. 46-53.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Sorting orders, subword complexes, Bruhat order and total positivity
AU - Armstrong, Drew
AU - Hersh, Patricia
PY - 2011/1
Y1 - 2011/1
N2 - In this note we construct a poset map from a Boolean algebra to the Bruhat order which unveils an interesting connection between subword complexes, sorting orders, and certain totally nonnegative spaces. This relationship gives a simple new proof that the proper part of Bruhat order is homotopy equivalent to the proper part of a Boolean algebra - that is, to a sphere. We also obtain a geometric interpretation for sorting orders. We conclude with two new results: that the intersection of all sorting orders is the weak order, and the union of sorting orders is the Bruhat order.
AB - In this note we construct a poset map from a Boolean algebra to the Bruhat order which unveils an interesting connection between subword complexes, sorting orders, and certain totally nonnegative spaces. This relationship gives a simple new proof that the proper part of Bruhat order is homotopy equivalent to the proper part of a Boolean algebra - that is, to a sphere. We also obtain a geometric interpretation for sorting orders. We conclude with two new results: that the intersection of all sorting orders is the weak order, and the union of sorting orders is the Bruhat order.
KW - Bruhat order
KW - Homotopy type
KW - Quillen Fiber Lemma
KW - Weak order
UR - http://www.scopus.com/inward/record.url?scp=79953699022&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=79953699022&partnerID=8YFLogxK
U2 - 10.1016/j.aam.2010.09.006
DO - 10.1016/j.aam.2010.09.006
M3 - Article
VL - 46
SP - 46
EP - 53
JO - Advances in Applied Mathematics
JF - Advances in Applied Mathematics
SN - 0196-8858
IS - 1-4
ER -