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University of Oregon

1.
Muth, Robert.
Representations of Khovanov-Lauda-Rouquier *algebras* of affine Lie type.

Degree: PhD, Department of Mathematics, 2016, University of Oregon

URL: http://hdl.handle.net/1794/20432

We study representations of Khovanov-Lauda-Rouquier (KLR) algebras of affine Lie type. Associated to every convex preorder on the set of positive roots is a system of cuspidal modules for the KLR algebra. For a balanced order, we study imaginary semicuspidal modules by means of `imaginary Schur-Weyl duality'. We then generalize this theory from balanced to arbitrary convex preorders for affine ADE types. Under the assumption that the characteristic of the ground field is greater than some explicit bound, we prove that KLR algebras are properly stratified. We introduce affine zigzag algebras and prove that these are Morita equivalent to arbitrary imaginary strata if the characteristic of the ground field is greater than the bound mentioned above. Finally, working in finite or affine affine type A, we show that skew Specht modules may be defined over the KLR algebra, and real cuspidal modules associated to a balanced convex preorder are skew Specht modules for certain explicit hook shapes.
*Advisors/Committee Members: Kleshchev, Alexander (advisor).*

Subjects/Keywords: KLR algebras; Representation theory

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APA (6^{th} Edition):

Muth, R. (2016). Representations of Khovanov-Lauda-Rouquier algebras of affine Lie type. (Doctoral Dissertation). University of Oregon. Retrieved from http://hdl.handle.net/1794/20432

Chicago Manual of Style (16^{th} Edition):

Muth, Robert. “Representations of Khovanov-Lauda-Rouquier algebras of affine Lie type.” 2016. Doctoral Dissertation, University of Oregon. Accessed November 28, 2020. http://hdl.handle.net/1794/20432.

MLA Handbook (7^{th} Edition):

Muth, Robert. “Representations of Khovanov-Lauda-Rouquier algebras of affine Lie type.” 2016. Web. 28 Nov 2020.

Vancouver:

Muth R. Representations of Khovanov-Lauda-Rouquier algebras of affine Lie type. [Internet] [Doctoral dissertation]. University of Oregon; 2016. [cited 2020 Nov 28]. Available from: http://hdl.handle.net/1794/20432.

Council of Science Editors:

Muth R. Representations of Khovanov-Lauda-Rouquier algebras of affine Lie type. [Doctoral Dissertation]. University of Oregon; 2016. Available from: http://hdl.handle.net/1794/20432

2.
Loubert, Joseph.
Affine Cellularity of Finite Type *KLR* *Algebras*, and Homomorphisms Between Specht Modules for *KLR* *Algebras* in Affine Type A.

Degree: PhD, Department of Mathematics, 2015, University of Oregon

URL: http://hdl.handle.net/1794/19255

This thesis consists of two parts. In the first we prove that the Khovanov-Lauda-Rouquier algebras R_α 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_α are generated by idempotents. This in particular implies the (known) result that the global dimension of R_α 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.
*Advisors/Committee Members: Kleshchev, Alexander (advisor).*

Subjects/Keywords: Affine cellularity; KLR algebras; Specht modules

…Cellularity of *KLR* *Algebras* of Finite Types
The content of chapter II has already been published as… …*KLR* *algebras* of finite types.
Our approach is independent of the homological results in… …the definition and basic results of KhovanovLauda-Rouquier (*KLR*) *algebras*. The… …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…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Loubert, J. (2015). Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A. (Doctoral Dissertation). University of Oregon. Retrieved from http://hdl.handle.net/1794/19255

Chicago Manual of Style (16^{th} Edition):

Loubert, Joseph. “Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A.” 2015. Doctoral Dissertation, University of Oregon. Accessed November 28, 2020. http://hdl.handle.net/1794/19255.

MLA Handbook (7^{th} Edition):

Loubert, Joseph. “Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A.” 2015. Web. 28 Nov 2020.

Vancouver:

Loubert J. Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A. [Internet] [Doctoral dissertation]. University of Oregon; 2015. [cited 2020 Nov 28]. Available from: http://hdl.handle.net/1794/19255.

Council of Science Editors:

Loubert J. Affine Cellularity of Finite Type KLR Algebras, and Homomorphisms Between Specht Modules for KLR Algebras in Affine Type A. [Doctoral Dissertation]. University of Oregon; 2015. Available from: http://hdl.handle.net/1794/19255