Degenerations of symplectic flag varieties via Lie theory and tropical geometry

• Degenerationen symplektischer Fahnenvarietäten durch Lie-Theorie und tropischer Geometrie

Balla, George; Fourier, Ghislain Paul Thomas (Thesis advisor); Markwig, Hannah (Thesis advisor); Littelmann, Peter (Thesis advisor)

Aachen : RWTH Aachen University (2023)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2023

Abstract

In this thesis, we study flat degenerations of projective varieties. Our methods stem from Lie theory, tropical geometry and combinatorics. More particularly, we consider symplectic flag varieties, which can be understood as homogeneous spaces corresponding to the symplectic Lie group. In the first part, we treat flat degenerations that arise by Lie theoretic methods. These are constructed by considering associated graded spaces corresponding to induced Poincaré-Birkoff-Witt (PBW) filtrations on simple highest weight modules for the symplectic Lie algebra. Such degenerations are called PBW degenerations. The corresponding symplectic degenerate flag varieties have been studied by Feigin, Finkelberg and Littelmann who have shown that these varieties are normal locally complete intersections with rational singularities and Frobenius split. Moreover, it has been shown that these varieties can be recovered as Schubert varieties in certain symplectic partial flag varieties for a symplectic group of higher rank. We consider the Plücker embedding of these varieties and give a full description for their defining ideal under this embedding. We use combinatorics of certain analogues of Young tableaux called PBW tableaux, which label a weighted basis of the homogeneous coordinate ring. It is not always true that the PBW degenerations are compatible with subvarieties. We fully characterize symplectic Schubert subvarieties that admit this compatibility in terms of symplectic Weyl group elements. In the second part, we use tropical methods to construct more general flat degenerations which specialize to the ones discussed above. These degenerations are labeled by a maximal prime cone of the tropical symplectic flag variety. The PBW tableaux are still a basis for the homogeneous coordinate ring in this setting and we use their combinatorics in our constructions. We show that every point in the relative interior of the above maximal cone corresponds to a flat degeneration into a toric variety associated with the Feigin-Fourier-Littelmann-Vinberg polytope. Along the way, we study in their own interest, tropical symplectic Grassmannians, which are building blocks for tropical symplectic flag varieties, and we provide a complete characterization.

Institutions

• Department of Mathematics [110000]
• Chair of Algebra and Representation Theory [114410]