This thesis consists of two parts. In the first we prove that the Khovanov-Lauda-Rouquier algebras $R_\alpha$ of finite type are (graded) affine cellular in the sense of Koenig and Xi. In fact, we establish a stronger property, namely that the affine cell ideals in $R_\alpha$ are generated by idempotents. This in particular implies the (known) result that the global dimension of $R_\alpha$ is finite. In the second part we use the presentation of the Specht modules given by Kleshchev-Mathas-Ram to derive results about Specht modules. In particular, we determine all homomorphisms from an arbitrary Specht module to a fixed Specht module corresponding to any hook partition. Along the way, we give a complete description of the action of the standard KLR generators on the hook Specht module. This work generalizes a result of James. This dissertation includes previously published coauthored material.

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 .

xii, 76 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We study the graded representation theory of the Iwahori-Hecke algebra, denoted by Hd , of the symmetric group over a field of characteristic zero at a root of unity. More specifically, we use graded Specht modules to calculate the graded decomposition numbers for Hd . The algorithm arrived at is the Lascoux-Leclerc-Thibon algorithm in disguise. Thus we interpret the algorithm in terms of graded representation theory. We then use the algorithm to compute several examples and to obtain a closed form for the graded decomposition numbers in the case of two-column partitions. In this case, we also precisely describe the 'reduction modulo p' process, which relates the graded irreducible representations of Hd over [Special characters omitted.] at a p th -root of unity to those of the group algebra of the symmetric group over a field of characteristic p. / Committee in charge: Alexander Kleshchev, Chairperson, Mathematics; Jonathan Brundan, Member, Mathematics; Boris Botvinnik, Member, Mathematics; Victor Ostrik, Member, Mathematics; William Harbaugh, Outside Member, Economics