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Classes of maximal-length reduced words in Coxeter groups

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Scott, Anthony (1996) Classes of maximal-length reduced words in Coxeter groups. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b1402578~S1

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Abstract

This thesis is concerned with the graph R of all reduced words for the longest element in the Coxeter groups of classical type, the edges representing braid relations.

In chapter 2 we consider the equivalence relation on the vertices of R generated by commuting adjacent letters if the corresponding simple reflections commute. An inductive way of describing all of the resulting commutation classes is described.

A characterisation of the quiver-compatible commutation classes, couched in terms of letter-multiplicities, is presented in chapter 3.

Chapter 4 introduces for each positive root ß an equivalence relation on R whose equivalence dasses are connected subgraphs called ß-components. It is shown that the ß-components are in bijective correspondence with the root vectors for ß (following Bedard) when ß is the highest root ∝0; in general there are more ß-components.

It happens that the natural quotient graph of ß-components is determined up to isomorphism by the length of ß; we choose to focus on the ∝0-components.

In chapter 5 we show that each ∝0-component in type A1, contains a unique quiver-compatible commutation class.

In chapter 6 we count the ∝0-components in type B1, by exhibiting explicit representatives which have a natural interpretation as partial quivers.

The edges of the graphs of ∝0-components in types A1, and B1, are determined by interpreting maximal chains in certain posets as elements of the Coxeter group of type A1-2 or B1-2, respectively.

Chapter 7 establishes an isomorphism between the graphs of ß-components in types B1 and D1 whenever ß is long.

Item Type: Thesis (PhD)
Subjects: Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH): Coxeter groups
Official Date: August 1996
Dates:
DateEvent
August 1996Submitted
Institution: University of Warwick
Theses Department: Mathematics Institute
Thesis Type: PhD
Publication Status: Unpublished
Supervisor(s)/Advisor: Carter, Roger W. (Roger William)
Sponsors: Engineering and Physical Sciences Research Council
Extent: 105 leaves
Language: eng

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