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.

This thesis is devoted to a problem in the representation theory of the symmetric group over C (the field of the complex numbers). Let d be a positive integer, and let S_d denote the symmetric group on d letters. Given a partition k of d, the Specht module V_k is a finite dimensional vector space over C which admits a natural basis indexed by all standard tableaux of shape k with entries in {1, 2, ..., d}. It affords an irreducible representation of the symmetric group S_d, and conversely every irreducible representation of S_d is isomorphic to V_k for some partition k. Given two Specht modules V_k, V_t their tensor product representation is in general reducible, and hence it splits into a direct sum of irreducibles. This raises the problem of describing the S_d equivariant projection morphisms (alternately called S_d-homomorphisms) in terms of the standard tableaux basis. In this work we give explicit formulae describing this morphism in the following cases: k=(d-1, 1), (d-2, 1,1), (2, 1,... ,1). Finally, we present a conjecture formula for the q-morphism in the case k=(d-r, 1, ..., 1).

This thesis is devoted to a problem in the representation theory of the symmetric group over C (the field of the complex numbers). Let d be a positive integer, and let S_d denote the symmetric group on d letters. Given a partition k of d, the Specht module V_k is a finite dimensional vector space over C which admits a natural basis indexed by all standard tableaux of shape k with entries in {1, 2, ..., d}. It affords an irreducible representation of the symmetric group S_d, and conversely every irreducible representation of S_d is isomorphic to V_k for some partition k. Given two Specht modules V_k, V_t their tensor product representation is in general reducible, and hence it splits into a direct sum of irreducibles. This raises the problem of describing the S_d equivariant projection morphisms (alternately called S_d-homomorphisms) in terms of the standard tableaux basis. In this work we give explicit formulae describing this morphism in the following cases: k=(d-1, 1), (d-2, 1,1), (2, 1,... ,1). Finally, we present a conjecture formula for the q-morphism in the case k=(d-r, 1, ..., 1).

Stuttgart, Univ., Diss., 2004.

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

Made available in DSpace on 2016-06-02T20:28:30Z (GMT). No. of bitstreams: 1 6214.pdf: 446462 bytes, checksum: 7f858711d074cd4b7beea958e52cd627 (MD5) Previous issue date: 2014-08-14 / Financiadora de Estudos e Projetos / Let F be a field of characteristic p ≥ 3. Denote by F0(X) and F1(X) the free associative algebras without and with unity, respectively, freely generated by the infinite set X. In this dissertation we study T-spaces and T-ideals infinitely generated. In [2] the mathematicians Aladova and Krasilnikov exhibited in F0(X) a T-ideal infinitely generated that contains the polynomial x2p. In [9], the mathematicians Goncalves, Krasilnikov and Sviridova exhibited in Fi(X) limits T-spaces when F is infinite. The objective of this dissertation is to study the results of the two papers cited above. / Seja F um corpo de característica p ≥ 3. Denote por F0(X) e F1(X) as algebras associativas livres sem e com unidade, respectivamente, livremente geradas pelo conjunto infinito X. Nesta dissertacão estudamos T-ideais e T-espacos infinitamente gerados. Em [2], os matematicos Aladova e Krasilnikov exibiram em F0(X) um T-ideal infinitamente gerado que contem o polinómio x2p. Em [9], os matematicos Goncalves, Krasilnikov e Sviridova exibiram em Fi(X) infinitos T-espacos limites quando F e infinito. O objetivo desta dissertaçao e estudar os resultados dos dois artigos citados acima.

In this dissertation we first study partition posets and their topology. For each composition c we show that the order complex of the poset of pointed set partitions is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module of a border strip associated to the composition. We also study the filter of pointed set partitions generated by knapsack integer partitions. In the second half of this dissertation we study descent avoidance in permutations. We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern, we show how to find the equation determining the spectrum. We give two length 4 applications, and a weighted pattern of length 3 where the associated operator only has one non-zero eigenvalue. Using generating functions we show that the error term in the asymptotic expression is the smallest possible.