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University of Oregon

1. Steinberg, David. Homological Properties of Standard KLR Modules.

Degree: PhD, Department of Mathematics, 2017, University of Oregon

URL: http://hdl.handle.net/1794/22292

Khovanov-Lauda-Rouquier algebras, or KLR algebras, are a family of algebras known to categorify the upper half of the quantized enveloping algebra of a given Lie algebra. In finite type, these algebras come with a family of standard modules, which correspond to PBW bases under this categorification. In this thesis, we show that there are no non-zero homomorphisms between distinct standard modules and that all non-zero endomorphisms of standard modules are injective. We then apply this result to obtain applications to the modular representation theory of KLR algebras. Restricting our attention to finite type A, we are then able to compute explicit projective resolutions of all standard modules. Finally, in finite type A when alpha is a positive root, we let D be the direct sum of all distinct standard modules and compute the algebra structure on Ext(D, D). This dissertation includes unpublished co-authored material.
*Advisors/Committee Members: Kleshchev, Alexander (advisor).*

Subjects/Keywords: Algebra; Category; KLR; Representation

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

APA (6^{th} Edition):

Steinberg, D. (2017). Homological Properties of Standard KLR Modules. (Doctoral Dissertation). University of Oregon. Retrieved from http://hdl.handle.net/1794/22292

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

Steinberg, David. “Homological Properties of Standard KLR Modules.” 2017. Doctoral Dissertation, University of Oregon. Accessed September 28, 2020. http://hdl.handle.net/1794/22292.

MLA Handbook (7^{th} Edition):

Steinberg, David. “Homological Properties of Standard KLR Modules.” 2017. Web. 28 Sep 2020.

Vancouver:

Steinberg D. Homological Properties of Standard KLR Modules. [Internet] [Doctoral dissertation]. University of Oregon; 2017. [cited 2020 Sep 28]. Available from: http://hdl.handle.net/1794/22292.

Council of Science Editors:

Steinberg D. Homological Properties of Standard KLR Modules. [Doctoral Dissertation]. University of Oregon; 2017. Available from: http://hdl.handle.net/1794/22292

University of Oregon

2. 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

Record Details Similar Records

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

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 September 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 Sep 2020.

Vancouver:

Muth R. Representations of Khovanov-Lauda-Rouquier algebras of affine Lie type. [Internet] [Doctoral dissertation]. University of Oregon; 2016. [cited 2020 Sep 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

3. 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

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 September 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 Sep 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 Sep 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