Algebra Seminar: Parkinson -- Cone types and automata, and regular partitions in Coxeter groups

James Parkinson (University of Sydney) 

Friday 20 May, 12-1pm, Place: Carslaw 273 

Title: Cone types and automata, and regular partitions in Coxeter groups.  

Abstract: In 1993 Brink and Howlett showed that finitely generated Coxeter groups are
automatic.  One ingredient was the construction of a finite state automaton recognising
the language of reduced words in the Coxeter group using the remarkable (finite!) set of
roots of "elementary roots" of the associated root system.  Recently Dehornoy, Dyer, and
Hohlweg introduced the notion of a Garside shadow in a Coxeter group, resulting in
further constructions of automata recognising the language of reduced words, and various
conjectures.  In this talk we outline recent joint work with Y.  Yau towards resolving
these conjectures.  This work centres around the notion of a "regular partition" of a
Coxeter group, and we show that such partitions are essentially equivalent to the class
of automata recognising the language of reduced words.  Moreover, we use this framework
to prove some fundamental facts about the minimal automata recognising the language of
reduced words, and study the associated "cone types" in Coxeter groups.