Sorting orders,subword complexes,Bruhat order and total positivity |
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Institution: | 1. Department of Mathematics, University of Miami, 1365 Memorial Drive, Ungar 515, Coral Gables, FL 33146, United States;2. Box 8205, North Carolina State University, Raleigh, NC 27695-8205, United States |
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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. |
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