Automatic Structures for Coxeter Groups
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Open Access
Type
ThesisThesis type
Doctor of PhilosophyAuthor/s
Yau, YeeAbstract
In this thesis, we study automatic structures recognising the language of reduced words for Coxeter groups. In particular, we focus on conditions for minimality for these automata. We give a classification of the Coxeter systems (W,S) for which the Brink-Howlett automaton is minimal ...
See moreIn this thesis, we study automatic structures recognising the language of reduced words for Coxeter groups. In particular, we focus on conditions for minimality for these automata. We give a classification of the Coxeter systems (W,S) for which the Brink-Howlett automaton is minimal in terms of excluded sub-graphs of the Coxeter graph ΓW , thereby resolving a conjecture of Hohlweg, Nadeau and Williams. We study the minimal automaton for W by studying the Cannon cone types of W. We investigate their basic properties and the partition of W induced by its cone types. Notably, a characterisation of cone types in terms of a minimal set of roots is given. Furthermore, it is proven that for every cone type of W, there is a unique representative of minimal length, which is a suffix of every element with the same cone type. We progress towards proving that the set of minimal length cone type representatives is a Garside shadow; which would prove that the automaton built from the smallest Garside shadow is minimal (also a conjecture of Hohlweg, Nadeau and Williams). We prove this for certain classes of Coxeter groups and initiate an approach to prove this in general.
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See moreIn this thesis, we study automatic structures recognising the language of reduced words for Coxeter groups. In particular, we focus on conditions for minimality for these automata. We give a classification of the Coxeter systems (W,S) for which the Brink-Howlett automaton is minimal in terms of excluded sub-graphs of the Coxeter graph ΓW , thereby resolving a conjecture of Hohlweg, Nadeau and Williams. We study the minimal automaton for W by studying the Cannon cone types of W. We investigate their basic properties and the partition of W induced by its cone types. Notably, a characterisation of cone types in terms of a minimal set of roots is given. Furthermore, it is proven that for every cone type of W, there is a unique representative of minimal length, which is a suffix of every element with the same cone type. We progress towards proving that the set of minimal length cone type representatives is a Garside shadow; which would prove that the automaton built from the smallest Garside shadow is minimal (also a conjecture of Hohlweg, Nadeau and Williams). We prove this for certain classes of Coxeter groups and initiate an approach to prove this in general.
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Date
2021Rights statement
The author retains copyright of this thesis. It may only be used for the purposes of research and study. It must not be used for any other purposes and may not be transmitted or shared with others without prior permission.Faculty/School
Faculty of Science, School of Mathematics and StatisticsAwarding institution
The University of SydneyShare