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The absolute order on the symmetric group, constructible partially ordered sets and Cohen-Macaulay complexes. (English) Zbl 1189.05173
Summary: The absolute order is a natural partial order on a Coxeter group \(W\). It can be viewed as an analogue of the weak order on \(W\) in which the role of the generating set of simple reflections in \(W\) is played by the set of all reflections in \(W\). By use of a notion of constructibility for partially ordered sets, it is proved that the absolute order on the symmetric group is homotopy Cohen-Macaulay. This answers in part a question raised by V. Reiner and the first author. The Euler characteristic of the order complex of the proper part of the absolute order on the symmetric group is also computed.

MSC:
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05A18 Partitions of sets
06A07 Combinatorics of partially ordered sets
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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