Ph.D. Thesis, Massachusetts Institute of Technology, 2006
This thesis examines several aspects of
reduced decompositions in finite Coxeter groups. Effort is primarily
concentrated on the symmetric group, although some discussions are
subsequently expanded to finite Coxeter groups of types B and
D.
In the symmetric group, the combined frameworks of permutation patterns and
reduced decompositions are used to prove a new characterization of vexillary
permutations. This characterization and the methods used yield a variety of new
results about the structure of several objects relating to a permutation.
These include its commutation classes, the corresponding graph of the classes,
the zonotopal tilings of a particular polygon, and a poset defined in terms of
these tilings. The class of freely braided permutations behaves particularly
well, and its graphs and posets are explicitly determined.
The Bruhat order for the symmetric group is examined, and the permutations with
boolean principal order ideals are completely characterized. These form an
order ideal which is a simplicial poset, and its rank generating function is
computed. Moreover, it is determined when the set of permutations avoiding a
particular set of patterns is an order ideal, and the rank generating functions
of these ideals are computed. The structure of the intervals and order ideals
in this poset is elucidated via patterns, including progress towards
understanding the relationship between pattern containment and subintervals in
principal order ideals.
The final discussions of the thesis are on reduced decompositions in the
finite Coxeter groups of types B and D. Reduced
decompositions of the longest element in the hyperoctahedral group are
studied, and expected values are calculated, expanding on previous work for
the symmetric group. These expected values give a quantitative interpretation
of the effects of the Coxeter relations on reduced decompositions of the
longest element in this group. Finally, the Bruhat order in types B
and D is studied, and the elements in these groups with boolean
principal order ideals are characterized and enumerated by length.