Geometric Invariant Theory and Applications to Tensor Products and the Saturation Conjecture
Public Deposited- Creator
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Kiers, Joshua
- College of Arts and Sciences, Department of Mathematics
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Kiers, Joshua
- Abstract
- Irreducible representations of a connected, semisimple, complex Lie group are parametrized by dominant, integral weights. If one such Lie group is embedded inside another, it is natural to ask when the tensor product of an irreducible representation of the smaller group with one of the larger group possesses a nontrivial space of invariants (this is equivalent to the representation theory branching problem). Geometric invariant theory is well-equipped to answer the slightly weaker question of existence of asymptotic invariants. After extensive work by several mathematicians, it is known that the solution set to this question is defined by finitely many rational linear inequalities and so induces a rational pointed cone. We provide formulas for the extremal rays of this cone by generalizing analogous formulas recently proven by the author and Belkale for the more specific scenario when the larger group is a direct product of the smaller group with itself (with the diagonal embedding). New rays not covered by these formulas also appear, but these are rare and easily identifiable by a procedure which we provide. We identify our first extremal rays thanks to a rigidity property stemming from a generalization of a conjecture of Fulton, which we first prove. Finally, we answer the Saturation Conjecture for three new connected, simple, complex Lie groups: type D_5, type D_6, and type E_6 (the first of the simply-laced exceptional types). This conjecture concerns the aforementioned cone in the diagonal embedding case and predicts that the need for weakening the original question to the asymptotic one is not necessary assuming the Lie group is simple of simply-laced type. Our verification takes as inspiration the proof of saturation for type D_4 given by Kapovich, Kumar, and Millson. However, we introduce a computational tool coming from the Localization Theorem in order to more efficiently calculate the necessary cup products using a supercomputer. Furthermore, the scale of the D_6 and E_6 calculations necessitates the use of the extremal ray formulas in order to directly produce rays instead of relying on software to find rays from inequalities.
- Date of publication
- 2020
- Keyword
- DOI
- Resource type
- Advisor
- Belkale, Prakash
- Kumar, Shrawan
- Rimanyi, Richard
- Hong, Jiuzu
- Thompson, Daniel
- Degree
- Doctor of Philosophy
- Degree granting institution
- University of North Carolina at Chapel Hill Graduate School
- Graduation year
- 2020
- Language
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