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1. Muthiah, Dinakar. Double MV Cycles, Affine PBW Bases, and Crystal Combinatorics.

Degree: PhD, Mathematics, 2013, Brown University

The theory of Mirkovic-Vilonen (MV) cycles and polytopes associated to a complex reductive group G has proven to be a rich source of structures related to representation theory. MV polytopes have proven to be a useful tool in understanding and unifying many constructions of crystals for finite-type Kac-Moody algebras. These polytopes arise naturally in many places, including the affine Grassmannian, pre-projective algebras, PBW bases, and KLR algebras. This thesis is part of an ongoing program to extend this theory to the affine Kac-Moody algebras In the first part of this thesis, we investigate double MV cycles, which are analogues of MV cycles in the case of an affine Kac-Moody group. We prove an explicit formula for the Braverman-Finkelberg-Gaitsgory crystal structure on double MV cycles, generalizing a finite-dimensional result of Baumann and Gaussent. As an application, we give a geometric construction of the Naito-Sagaki-Saito crystal via the action of affine SL_n on Fermionic Fock space. In particular, this construction gives rise to an isomorphism of crystals between the set of double MV cycles and the Naito-Sagaki-Saito crystal. As a result, we can independently prove that the Naito-Sagaki-Saito crystal is the B-infinity crystal. In particular, our geometric proof works in the previously unknown case of affine SL_2. In the second part of this thesis, which is joint work with Peter Tingley, we investigate the recently proposed notions of affine MV polytopes. A definition of MV polytopes in symmetric affine cases has been proposed using pre-projective algebras. In the rank-2 affine cases, a combinatorial definition has also been proposed. Additionally, the theory of PBW bases has been extended to affine cases, and, at least in rank-2, we show that this can also be used to define MV polytopes. The main result of this paper is that these three notions of MV polytope all agree in the relevant rank-2 cases. As a corollary, we can give a complete combinatorial characterization of the affine MV polytopes arising from pre-projective algebras. Our main tool is a new characterization of rank-2 affine MV polytopes. Advisors/Committee Members: Braverman, Alexander (Director), Abramovich, Dan (Reader), Harris, Bruno (Reader).

Subjects/Keywords: affine Lie algebras

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

APA (6th Edition):

Muthiah, D. (2013). Double MV Cycles, Affine PBW Bases, and Crystal Combinatorics. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:320617/

Chicago Manual of Style (16th Edition):

Muthiah, Dinakar. “Double MV Cycles, Affine PBW Bases, and Crystal Combinatorics.” 2013. Doctoral Dissertation, Brown University. Accessed September 20, 2020. https://repository.library.brown.edu/studio/item/bdr:320617/.

MLA Handbook (7th Edition):

Muthiah, Dinakar. “Double MV Cycles, Affine PBW Bases, and Crystal Combinatorics.” 2013. Web. 20 Sep 2020.

Vancouver:

Muthiah D. Double MV Cycles, Affine PBW Bases, and Crystal Combinatorics. [Internet] [Doctoral dissertation]. Brown University; 2013. [cited 2020 Sep 20]. Available from: https://repository.library.brown.edu/studio/item/bdr:320617/.

Council of Science Editors:

Muthiah D. Double MV Cycles, Affine PBW Bases, and Crystal Combinatorics. [Doctoral Dissertation]. Brown University; 2013. Available from: https://repository.library.brown.edu/studio/item/bdr:320617/

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