Mathematics Theses and Dissertations

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https://hdl.handle.net/1794/2044

This collection contains some of the theses and dissertations produced by students in the University of Oregon Mathematics Graduate Program. Paper copies of these and other dissertations and theses are available through the UO Libraries.

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Browsing Mathematics Theses and Dissertations by Subject "Algebra"

Now showing 1 - 3 of 3
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    A SPECIAL ENDOMORPHISM OF THE STANDARD GAITSGORY CENTRAL OBJECT OF THE AFFINE HECKE CATEGORY
    (University of Oregon, 2024-03-25) Hathaway, Jay; Elias, Ben
    Using the combinatorial description of the standard Gaitsgory centralobject of the (extended, graded) affine type A Hecke category due to Elias, we show the existence of and explicitly describe the unique endomorphism that lifts right multiplication by the i-th fundamental weight on the i-th component of the associated graded of its Wakimoto filtration. We give work in progress on describing a conjectural program to categorify the Vershik-Okounkov approach to the representation theory of the affine Hecke algebra. Here this endomorphism will play a role. This is the affine version of the program described by Gorsky, Negut, and Rasmussen in finite type A.
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    Homological Properties of Standard KLR Modules
    (University of Oregon, 2017-05-01) Steinberg, David; Kleshchev, Alexander
    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 This dissertation includes unpublished co-authored material.
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    Modules with Good Filtrations over Generalized Schur Algebras
    (University of Oregon, 2022-10-04) Weinschelbaum, Ilan; Kleshchev, Alexander
    In this dissertation we examine generalized Schur algebras, as defined by Kleshchev and Muth. Given a quasi-hereditary superalgebra $A$, Kleshchev and Muth proved that for $n \geq d$, the generalized Schur algebra $T^A (n,d)$ is again quasi-hereditary.They described the bisuperalgebra struture on $T^A(n) := \bigoplus_d T^A(n,d)$. In particular, there is a coproduct which gives us a way to take a $T^A(n,d)$-module $V$ and $T^A(n,r)$-module $W$ and produce a $T^A(n,d+r)$-module $V \otimes W$. We will prove that if $V$ and $W$ each have standard (resp. costandard) filtrations, then so does $V \otimes W$. In the last chapter we will use this result to prove that in the case that $A$ is the extended zigzag algebra $\EZig$, the extended zigzag Schur algebra $T^\EZig(n,d)$ is Ringel self-dual for all $n \geq d$.