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University of Colorado

1. Ly, Megan Danielle. Schur – Weyl Duality for Unipotent Upper Triangular Matrices.

Degree: PhD, 2018, University of Colorado

URL: https://scholar.colorado.edu/math_gradetds/59

Schur – Weyl duality is a fundamental framework in combinatorial representation theory. It intimately relates the irreducible representations of a group to the irreducible representations of its centralizer algebra. We investigate the analogue of Schur – Weyl duality for the group of unipotent upper triangular matrices over a finite field. In this case, the character theory of these upper triangular matrices is "wild" or unattainable. Thus we employ a generalization, known as supercharacter theory, that creates a striking variation on the character theory of the symmetric group with combinatorics built from set partitions. In this thesis, we present a combinatorial formula for calculating a restriction and induction of supercharacters based on statistics of set partitions and seashell inspired diagrams. We use these formulas to create a graph that encodes the decomposition of a tensor space, and develop an analogue of Young tableaux, known as shell tableaux, to index paths in this graph. These paths also help determine a basis for the maps that centralize the action of the group of unipotent upper triangular matrices. We construct a part of this basis by determining copies of certain modules inside a tensor space to construct projection maps onto supermodules that act on a standard basis.
*Advisors/Committee Members: Nathaniel Thiem, Richard M. Green, Martin Walter, Amanda Schaeffer Fry, Farid Aliniaeifard.*

Subjects/Keywords: supercharacter; schur-weyl duality; matrices; theory; combinatiorial; Mathematics; Statistical Theory

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

APA (6^{th} Edition):

Ly, M. D. (2018). Schur – Weyl Duality for Unipotent Upper Triangular Matrices. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/59

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

Ly, Megan Danielle. “Schur – Weyl Duality for Unipotent Upper Triangular Matrices.” 2018. Doctoral Dissertation, University of Colorado. Accessed November 25, 2020. https://scholar.colorado.edu/math_gradetds/59.

MLA Handbook (7^{th} Edition):

Ly, Megan Danielle. “Schur – Weyl Duality for Unipotent Upper Triangular Matrices.” 2018. Web. 25 Nov 2020.

Vancouver:

Ly MD. Schur – Weyl Duality for Unipotent Upper Triangular Matrices. [Internet] [Doctoral dissertation]. University of Colorado; 2018. [cited 2020 Nov 25]. Available from: https://scholar.colorado.edu/math_gradetds/59.

Council of Science Editors:

Ly MD. Schur – Weyl Duality for Unipotent Upper Triangular Matrices. [Doctoral Dissertation]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/math_gradetds/59

University of Colorado

2. Shannon, Erica Hilary. Computing Invariant Forms for Lie Algebras Using Heaps.

Degree: PhD, Mathematics, 2016, University of Colorado

URL: https://scholar.colorado.edu/math_gradetds/46

In this thesis, I present a combinatorial formula for a symmetric invariant quartic form on a spin module for the simple Lie algebra d6. This formula relies on a description of this spin module as a vector space with weights, and weight vectors, indexed by ideals of a particular heap. I describe a new statistic, the profile, on pairs of heap ideals. The profile efficiently encodes the shape of the symmetric difference between the two ideals and demonstrates the available actions of the Weyl group and Lie algebra on any given pair. From the profile, I identify a property called a crossing. The actions of the Weyl group and Lie algebra on pairs of weights may be interpreted as adding or removing crossings between the corresponding ideals. Using the crossings, I present a formula for the symmetric invariant quartic form on a spin module for d6, and discuss potential applications to other closely related minuscule representations.
*Advisors/Committee Members: Richard M. Green, Nathaniel Thiem, Eric Stade, Martin Walter, Amanda Schaeffer Fry.*

Subjects/Keywords: Mathematics

Record Details Similar Records

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

APA (6^{th} Edition):

Shannon, E. H. (2016). Computing Invariant Forms for Lie Algebras Using Heaps. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/46

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

Shannon, Erica Hilary. “Computing Invariant Forms for Lie Algebras Using Heaps.” 2016. Doctoral Dissertation, University of Colorado. Accessed November 25, 2020. https://scholar.colorado.edu/math_gradetds/46.

MLA Handbook (7^{th} Edition):

Shannon, Erica Hilary. “Computing Invariant Forms for Lie Algebras Using Heaps.” 2016. Web. 25 Nov 2020.

Vancouver:

Shannon EH. Computing Invariant Forms for Lie Algebras Using Heaps. [Internet] [Doctoral dissertation]. University of Colorado; 2016. [cited 2020 Nov 25]. Available from: https://scholar.colorado.edu/math_gradetds/46.

Council of Science Editors:

Shannon EH. Computing Invariant Forms for Lie Algebras Using Heaps. [Doctoral Dissertation]. University of Colorado; 2016. Available from: https://scholar.colorado.edu/math_gradetds/46