57,649 research outputs found

    Quasi-Minuscule Quotients and Reduced Words for Reflections

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    We study the reduced expressions for reflections in Coxeter groups, with particular emphasis on finite Weyl groups. For example, the number of reduced expressions for any reflection can be expressed as the sum of the squares of the number of reduced expressions for certain elements naturally associated to the reflection. In the case of the longest reflection in a Weyl group, we use a theorem of Dale Peterson to provide an explicit formula for the number of reduced expressions. We also show that the reduced expressions for any Weyl group reflection are in bijection with the linear extensions of a natural partial ordering of a subset of the positive roots or co-roots.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46149/1/10801_2004_Article_333190.pd

    A scattering of orders

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    A linear ordering is scattered if it does not contain a copy of the rationals. Hausdorff characterised the class of scattered linear orderings as the least family of linear orderings that includes the class B \mathcal B of well-orderings and reversed well-orderings, and is closed under lexicographic sums with index set in B \mathcal B. More generally, we say that a partial ordering is Îș \kappa -scattered if it does not contain a copy of any Îș \kappa -dense linear ordering. We prove analogues of Hausdorff's result for Îș \kappa -scattered linear orderings, and for Îș \kappa -scattered partial orderings satisfying the finite antichain condition. We also study the QÎș \mathbb{Q}_\kappa -scattered partial orderings, where QÎș \mathbb{Q}_\kappa is the saturated linear ordering of cardinality Îș \kappa , and a partial ordering is QÎș \mathbb{Q}_\kappa -scattered when it embeds no copy of QÎș \mathbb{Q}_\kappa . We classify the QÎș \mathbb{Q}_\kappa -scattered partial orderings with the finite antichain condition relative to the QÎș \mathbb{Q}_\kappa -scattered linear orderings. We show that in general the property of being a QÎș \mathbb{Q}_\kappa -scattered linear ordering is not absolute, and argue that this makes a classification theorem for such orderings hard to achieve without extra set-theoretic assumptions
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