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Full-Text Articles in Physical Sciences and Mathematics
On The Cyclically Fully Commutative Elements Of Coxeter Groups, T. Boothby, J. Burket, M. Eichwald, D. C. Ernst, R. M. Green, Matthew Macauley
On The Cyclically Fully Commutative Elements Of Coxeter Groups, T. Boothby, J. Burket, M. Eichwald, D. C. Ernst, R. M. Green, Matthew Macauley
Publications
Let W be an arbitrary Coxeter group. If two elements have expressions that are cyclic shifts of each other (as words), then they are conjugate (as group elements) in W. We say that w is cyclically fully commutative (CFC) if every cyclic shift of any reduced expression for w is fully commutative (i.e., avoids long braid relations). These generalize Coxeter elements in that their reduced expressions can be described combinatorially by acyclic directed graphs, and cyclically shifting corresponds to source-to-sink conversions. In this paper, we explore the combinatorics of the CFC elements and enumerate them in all Coxeter groups. …
On Enumeration Of Conjugacy Classes Of Coxeter Elements, Matthew Macauley, Henning S. Mortveit
On Enumeration Of Conjugacy Classes Of Coxeter Elements, Matthew Macauley, Henning S. Mortveit
Publications
In this paper we study the equivalence relation on the set of acyclic orientations of a graph Y that arises through source-to-sink conversions. This source-to-sink conversion encodes, e.g. conjugation of Coxeter elements of a Coxeter group. We give a direct proof of a recursion for the number of equivalence classes of this relation for an arbitrary graph Y using edge deletion and edge contraction of non-bridge edges. We conclude by showing how this result may also be obtained through an evaluation of the Tutte polynomial as T (Y, 1, 0), and we provide bijections to two other classes of acyclic …