This thesis presents the results of an investigation into the representation theory of the alternating and dihedral groups and explores how their irreducible representations can be distinguished with the use of class sums.

In this thesis, we present a new class of algorithms that determine fast Fourier transforms for a given finite group G. These algorithms use eigenspace projections determined by a chain of subgroups of G, and rely on a path algebraic approach to the representation theory of finite groups developed by Ram (26). Applying this framework to the symmetric group, Sn, yields a class of fast Fourier transforms that we conjecture to run in O(n2n!) time. We also discuss several future directions for this research.

A discrete Fourier transform, or DFT, is an isomorphism from a group algebra to a direct sum of matrix algebras. An algorithm that efficiently applies a DFT is called a fast Fourier transform, or FFT. The concept of a DFT will be introduced and examined from both a general and algebraic perspective. We will then present and analyze a specific FFT for the symmetric group.

A Discrete Fourier Transform (DFT) changes the basis of a group algebra from the standard basis to a Fourier basis. An efficient application of a DFT is called a Fast Fourier Transform (FFT). This research pertains to a particular type of FFT called Decimation in Frequency (DIF). An efficient DIF has been established for commutative algebra; however, a successful analogue for non-commutative algebra has not been derived. However, we currently have a promising DIF algorithm for CSn called Orrison-DIF (ODIF). In this paper, I will formally introduce the ODIF and establish a bound on the operation count of the algorithm.

The regular representation of the symmetric group Sn is a vector space of dimension n! with many interesting invariant subspaces. The projections of a vector onto these subspaces may be computed by first considering projections onto certain basis elements in the subspace and then recombining later. If all of these projections are kept, it creates an explosion in the size of the data, making it difficult to store and work with. This is a study of techniques to compress this computed data such that it is of the same dimmension as the original vector.

We consider the problem of counting <i>transitive factorizations</i> of permutations; that is, we study tuples (&sigma;<i>r</i>,. . . ,&sigma;1) of permutations on {1,. . . ,<i>n</i>} such that (1) the product &sigma;<i>r</i>. . . &sigma;1 is equal to a given target permutation &pi;, and (2) the group generated by the factors &sigma;<i>i</i> acts transitively on {1,. . . ,<i>n</i>}. This problem is widely known as the <i>Hurwitz Enumeration Problem</i>, since an encoding due to Hurwitz shows it to be equivalent to the enumeration of connected branched coverings of the sphere by a surface of given genus with specified branching. Much of our work concerns the enumeration of transitive factorizations of permutations into a minimal number of transposition factors. This problem has received considerable attention, and a formula for the number <i>c</i>(&pi;) of such factorizations of an arbitrary permutation &pi; has been derived through various means. The formula is remarkably simple, being a product of well-known combinatorial numbers, but no bijective proof of it is known except in the special case where &pi; is a full cycle. A major goal of this thesis is to provide further combinatorial rationale for this formula. We begin by introducing an encoding of factorizations (into transpositions) as edge-labelled maps. Our central result is a bijection that allows trees to be "pruned" from such maps. This is shown to explain the appearance of factors of the form <i>k^k</i> in the aforementioned formula for <i>c</i>(&pi;). It also has the effect of shifting focus to the combinatorics of smooth maps (<i>i. e. </i> maps without vertices of degree one). By providing decompositions for certain smooth planar maps, we are able to give combinatorial evaluations of <i>c</i>(&pi;) when &pi; is composed of up to three cycles. Many of these results are generalized to factorizations in which the factors are cycles of any length. We also investigate the <i>Double Hurwitz Problem</i>, which calls for the enumeration of factorizations whose leftmost factor is of specified cycle type, and whose remaining factors are transpositions. Finally, we extend our methods to the enumeration of factorizations up to an equivalence relation induced by possible commutations between adjacent factors.

We consider the problem of counting <i>transitive factorizations</i> of permutations; that is, we study tuples (&sigma;<i>r</i>,. . . ,&sigma;1) of permutations on {1,. . . ,<i>n</i>} such that (1) the product &sigma;<i>r</i>. . . &sigma;1 is equal to a given target permutation &pi;, and (2) the group generated by the factors &sigma;<i>i</i> acts transitively on {1,. . . ,<i>n</i>}. This problem is widely known as the <i>Hurwitz Enumeration Problem</i>, since an encoding due to Hurwitz shows it to be equivalent to the enumeration of connected branched coverings of the sphere by a surface of given genus with specified branching. Much of our work concerns the enumeration of transitive factorizations of permutations into a minimal number of transposition factors. This problem has received considerable attention, and a formula for the number <i>c</i>(&pi;) of such factorizations of an arbitrary permutation &pi; has been derived through various means. The formula is remarkably simple, being a product of well-known combinatorial numbers, but no bijective proof of it is known except in the special case where &pi; is a full cycle. A major goal of this thesis is to provide further combinatorial rationale for this formula. We begin by introducing an encoding of factorizations (into transpositions) as edge-labelled maps. Our central result is a bijection that allows trees to be "pruned" from such maps. This is shown to explain the appearance of factors of the form <i>k^k</i> in the aforementioned formula for <i>c</i>(&pi;). It also has the effect of shifting focus to the combinatorics of smooth maps (<i>i. e. </i> maps without vertices of degree one). By providing decompositions for certain smooth planar maps, we are able to give combinatorial evaluations of <i>c</i>(&pi;) when &pi; is composed of up to three cycles. Many of these results are generalized to factorizations in which the factors are cycles of any length. We also investigate the <i>Double Hurwitz Problem</i>, which calls for the enumeration of factorizations whose leftmost factor is of specified cycle type, and whose remaining factors are transpositions. Finally, we extend our methods to the enumeration of factorizations up to an equivalence relation induced by possible commutations between adjacent factors.

No description available.

Let G be a finite group and X a subset of G. The commuting graph C(G,X) is the graph whose vertex set is X with two distinct elements of X joined by an edge whenever they commute in the group G. This thesis studies the structure of commuting graphs C(G,X) when G is either a symmetric group Sym(n) or a sporadic group McL, and X a conjugacy class for elements of order three. We describe how this graph can be useful in understanding various aspects of the structure of the group with a particular emphasis on the connectivity of the graph, the properties of the discs around some fixed vertex and the diameter of the graph.

Representation theory is the study of abstract algebraic structures by representing their elements as linear transformations or matrices. It provides a bridge between the abstract symbolic mathematics and its explicit applications in nearly every branch of mathematics. Combinatorial representation theory aims to use combinatorial objects to model representations, thus answering questions in this field combinatorially. Combinatorial objects are used to help describe, count and generate representations. This has led to a rich symbiotic relationship where combinatorics has helped answer algebraic questions and algebraic techniques have helped answer combinatorial questions. In this thesis we discuss the representation theory of the symmetric group and the general linear group. The theory of these two families of groups is often considered the corner stone of combinatorial representation theory. Results and techniques arising from the study of these groups have been successfully generalized to a very wide class of groups. An overview of some of the generalizations can be found in [BR99]. There are also many avenues for further generalizations which are currently being explored. The constructions of the Specht and Schur modules that we discuss here use the concept of Young tableaux. Young tableaux are combinatorial objects that were introduced by the Reverend Alfred Young, a mathematician at Cambridge University, in 1901. In 1903, Georg Frobenius applied them to the study of the symmetric group. Since then, they have been found to play an important role in the study of symmetric functions, representation theory of the symmetric and complex general linear groups and Schubert calculus of Grassmannians. Applications of Young tableaux to other branches of mathematics are still being discovered. When drawing and labelling Young tableaux there are a few conflicting conventions in the literature, throughout this thesis we shall be following the English notation. In chapter 1 we shall make a few definitions and state some results which will be used in this thesis. In chapter 2 we discuss the representations of the symmetric group. In this chapter we define the Specht modules and prove that they describe all the irreducible representations of Sn. We conclude with a discussion about the ring of Sn representations which is used to prove some identities of Specht modules. In chapter 3 we discuss the representations of the general linear group. In this chapter we define the Schur modules and prove that they describe all the irreducible rational representations of GLmC. We also show that the set of tableaux forms an indexing set for a basis of the Schur modules. In chapter 4 we describe a relation between the Specht and Schur modules. This is a corollary to the more general Schur-Weyl duality, an overview of which can be found in [BR99]. The appendix contains the code and screen-shots of two computer programs that were written as part of this thesis. The programs have been written in C++ and the data structures have been implemented using the Standard Template Library. The first program gives us information about the representations of Sn for a given n. For a user defined n it will list all the Specht modules corresponding to that n, their dimensions and the standard tableaux corresponding to their basis elements. The second program gives information about a certain representation of GLmC. For a user defined m and λ it gives the dimension and the semistandard tableaux corresponding to the basis elements of the Schur module Eλ .