Open Access Repository

The planar modular partition monoid

Downloads

Downloads per month over past year

Ham, NC (2016) The planar modular partition monoid. PhD thesis, University of Tasmania.

[img]
Preview
PDF (Whole thesis)
Ham_whole_thesi...pdf | Download (1MB)
Available under University of Tasmania Standard License.

| Preview

Abstract

The primary contribution of this thesis is to introduce and examine the planar modular partition monoid
for parameters m, k ∈ Z>0, which has simultaneously and independently generated interest from other
researchers as outlined within.
Our collective understanding of related monoids, in particular the Jones, Brauer and partition monoids,
along with the algebras they generate, has heavily infuenced the direction of research by a significant
number of mathematicians and physicists. Examples include Schur-Weyl type dualities in representation
theory along with Potts, ice-type and Andrew-Baxter-Forrester models from statistical mechanics, giving
strong motivation for the planar modular partition monoid to be examined.
The original results contained within this thesis relating to the planar modular partition monoid are: the
establishment of generators; recurrence relations for the cardinality of the monoid; recurrence relations
for the cardinality of Green's R, L and D relations; and a conjecture on relations that appear to present
the planar modular partition monoid when m = 2. For diagram semigroups that are closed under vertical
reflections, characterisations of Green's R, L and H relations have previously been established using the
upper and lower patterns of bipartitions. We give a characterisation of Green's D relation with a similar
flavour for diagram semigroups that are closed under vertical reflections.
We also give a number of analogous results for the modular partition monoid, the monoid generated by
replacing diapses with m-apses in the generators of the Jones monoid, later referred to as the m-apsis
monoid, and the join of the m-apsis monoid with the symmetric group.
A further contribution of this thesis is a reasonably comprehensive exposition of the fundamentals of
diagram semigroups, which have traditionally been approached from the representation theory side and
have since blossomed into a thriving area of research in their own right.

Item Type: Thesis (PhD)
Keywords: Algebra, Combinatorics, Discrete Mathematics
Copyright Information:

Copyright 2015 the author

Date Deposited: 18 Oct 2016 01:45
Last Modified: 18 Oct 2016 01:45
Item Statistics: View statistics for this item

Actions (login required)

Item Control Page Item Control Page
TOP