The hook fusion procedure for Hecke algebras

We derive a new expression for the q-analogue of the Young symmetrizer which generate irreducible representations of the Hecke algebra. We obtain this new expression using Cherednik's fusion procedure. However, instead of splitting Young diagrams into their rows or columns, we consider their principal hooks. This minimises the number of auxiliary parameters needed in the fusion procedure.Comment: 19 page

Adjoint functors and triangulated categories

We give a construction of triangulated categories as quotients of exact categories where the subclass of objects sent to zero is defined by a triple of functors. This includes the cases of homotopy and stable module categories. These categories naturally fit into a framework of relative derived categories, and once we prove that there are decent resolutions of complexes, we are able to prove many familiar results in homological algebra.Comment: 20 pages. Added subsection on Transfer in the homotopy category, corrected typos (May 2006). Expanded to 20 pages, minor corrections, accepted for publication in Comm.in Alg. (August 2007

The new external microbeam facility of the Oxford nuclear microprobe and its application to problems in archaeological science

Recent developments of the extemal beam facility of the Oxford Nuclear Microprobe have led to an enhancement of the capabilities of the instrument for analysing large or sensitive objects in air with a spatial resolution of 50 to 100 (xm. Used in conjunction with the 1 pm resolution in-vacuo facility this provides a unique elemental analysis facility which is being applied to a number of archaeological problems. This paper describes briefly the capabilities of the faci lity and outlines the range of applications

The hook fusion procedure and its generalisations

The fusion procedure provides a way to construct new solutions to the Yang-Baxter equation. In the case of the symmetric group the fusion procedure has been used to construct diagonal matrix elements using a decomposition of the Young diagram into its rows or columns. We present a new construction which decomposes the diagram into hooks, the great advantage of this is that it minimises the number of auxiliary parameters needed in the procedure. We go on to use the hook fusion procedure to find diagonal matrix elements computationally and calculate supporting evidence to a previous conjecture. We are motivated by the construction of certain elements that allow us to generate representations of the symmetric group and single out particular irreducible components. In this way we may construct higher representations of the symmetric group from elementary ones. We go some way to generalising the hook fusion procedure by considering other decompositions of Young diagrams, specifically into ribbons. Finally, we adapt our construction to the quantum deformation of the symmetric group algebra known as the Hecke algebra.Comment: Thesis 135 page

Fitting ideals and module structure

Let R be a commutative ring with a 1. Original work by H. Fitting showed how we can associate to each finitely generated E-module a unique sequence of R-ideals, which are known as Fitting Ideals. The aim of this thesis is to undertake an investigation of Fitting Ideals and their relation with module structure and to construct a notion of Fitting Invariant for certain non-commutative rings. We first of all consider the commutative case and see how Fitting Ideals arise by considering determinantal ideals of presentation matrices of the underlying module and we describe some applications. We then study the behaviour of Fitting Ideals for certain module structures and investigate how useful Fitting Ideals are in determining the underlying module. The main part of this work considers the non-commutative case and constructs Fitting Invariants for modules over hereditary orders and shows how, by considering maximal orders and projectives in the hereditary order, we can obtain some very useful invariants which ultimately determine the structure of torsion modules. We then consider what we can do in the non-hereditary case, in particular for twisted group rings. Here we construct invariants by adjusting presentation matrices which generalises the previous work done in the hereditary case