Sorting orders, subword complexes, Bruhat order and total positivity

Details Sorting orders, subword complexes, Bruhat order and total positivity Sorting orders, subword complexes, Bruhat order and total positivity

2009-09-15

arxiv.org/abs/0909.2828v2

Mathematics

Combinatorics

10 pages, 1 figure. This is the official version. It is more official than the version that appears in Advances in Applied Mat

Scientific paper

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 new proof of Bj\"orner and Wachs' result \cite{BW} 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.

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