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Title Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A
Publication Date
Date Accessioned
Degree PhD
Discipline/Department Department of Mathematics
Degree Level doctoral
University/Publisher University of Oregon
Abstract 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.
Subjects/Keywords Affine cellularity; KLR algebras; Specht modules
Contributors Kleshchev, Alexander (advisor)
Language en
Rights Creative Commons BY 4.0-US [Always confirm rights and permissions with the source record.]
Country of Publication us
Record ID handle:1794/19255
Repository oregon
Date Retrieved
Date Indexed 2020-06-18
Grantor University of Oregon
Issued Date 2015-08-18 00:00:00

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…and Ram (20) provide a presentation for the Specht modules as modules over the full KLR algebra R, thus redefining these classical objects purely in the context of the KLR algebras. Our second main theorem determines Hom(S µ , S λ…

…x29; when λ is a hook. Affine Cellularity of KLR Algebras of Finite Types The content of chapter II has already been published as (18). The goal of chapter II is to establish (graded) affine cellularity in the sense of Koenig and Xi…

…graded) cellularity of cyclotomic KLR algebras of finite types. Our approach is independent of the homological results in (30), (13) and (3) (which relies on (30)). The connection between the theory…

…theoretic notation that we employ. We move on in subsection 2.1 to the definition and basic results of KhovanovLauda-Rouquier (KLR) algebras. The next two subsections are devoted to recalling results about the representation theory of KLR algebras

…Then, in subsection 2.1, we introduce our notation regarding quantum groups, and recall some well-known basis theorems. The next subsection is devoted to the connection between KLR algebras and quantum groups, namely the categorification theorems…

…Finally, subsection 2.1 contains an easy direct proof of a graded dimension formula for the KLR algebras, cf. (3, Corollary 3.15). 6 Section 3 is devoted to constructing a basis for the KLR algebras that is amenable to checking affine…

…basis in full generality. In section 4 we show how the affine cellular basis is used to prove that the KLR algebras are affine cellular. Finally, in section 5 we verify Hypothesis 2.2.9 for all positive roots in all finite types. We begin in subsection…

…RdΛ0 known as a cyclotomic KLR (1) (Khovanov-Lauda-Rouquier) algebra of type Ae−1 when e 6= 0, and type A∞ when e = 0. The Specht modules are described in (4) as modules over the cyclotomic KLR algebras. This result is…