arXiv subjects

George Balla

Publications and source records attributed to George Balla.

6 recordsLinked to original sources

arXiv2025

Shellings of Unbounded Polyhedra

The shellability of the boundary complex of an unbounded polyhedron is investigated. To this end, it is necessary to study suitable compactifications first. Results on polyhedra can then be exploited to derive a shellability result for tropical hypersurfaces. Under the hood there is a subtle interplay between the duality of polyhedral complexes and their shellability. Translated into discrete Morse theory, that interplay also gives that the tight span of a regular subdivision is collapsible, but not shellable in general.

math.CO↗

arXiv2024

Regular subdivisions, bounds on initial ideals, and categorical limits

Several known constructions relate initial degenerations of projective toric varieties and Grassmannians to regular subdivisions of appropriate point configurations. We define a general framework which allows for partial generalizations of these constructions to arbitrary projective schemes (as well as their very affine parts). We associate a point configuration $A$ with any homogeneous ideal $I$. We obtain upper and lower bounds on every initial ideal of $I$, defining them in terms of the regular subdivision of $A$ given by the same weight. Furthermore, both bounds are interpreted categorically via (co)limits over the face poset of the subdivision. We also investigate when these bounds are exact, showing that the respective weights form a subfan in the secondary fan of $A$.

math.CO↗

arXiv2023

Tropical symplectic flag varieties: a Lie-theoretic approach

We study tropicalization of symplectic flag varieties with respect to the Pl\"ucker embedding. We identify a particular maximal prime cone in this tropicalization by explicitly giving its facets. For every interior point of this maximal cone, the corresponding Gr\"obner degeneration is the toric variety associated to the Feigin-Fourier-Littelmann-Vinberg (FFLV) polytope. Our main tool is the notion of birational sequences introduced by Fourier, Littelmann and the second author, which bridges between weighted PBW filtrations of representations of symplectic Lie algebras and degree functions on defining ideals of symplectic flag varieties.

math.AG↗

arXiv2022

PBW filtration and monomial bases for Demazure modules in types A and C

We characterise the symplectic Weyl group elements such that the FFLV basis is compatible with the PBW filtration on symplectic Demazure modules, extending type A results by the second author. Surprisingly, the number of such elements depends not on the type A or C of the Lie algebra but on the rank only.

math.RT↗

arXiv2022

Symplectic PBW Degenerate Flag Varieties; PBW Tableaux and Defining Equations

We define a set of PBW-semistandard tableaux that is in a weight preserving bijection with the set of monomials corresponding to integral points in the Feigin-Fourier-Littelmann-Vinberg polytope for highest weight modules of the symplectic Lie algebra. We then show that these tableaux parametrize bases of the homogeneous coordinate rings of the complete symplectic original and PBW degenerate flag varieties. From this construction, we provide explicit degenerate relations that generate the defining ideal of the PBW degenerate variety. These relations consist of type A degenerate Plücker relations and a set of degenerate linear relations that we obtain from De Concini's linear relations.

math.RT↗

arXiv2021

The tropical symplectic Grassmannian

We launch the study of the tropicalization of the symplectic Grassmannian, that is, the space of all linear subspaces that are isotropic with respect to a fixed symplectic form. We formulate tropical analogues of several equivalent characterizations of the symplectic Grassmannian and determine all implications between them. In the process, we show that the Plücker and symplectic relations form a tropical basis if and only if the rank is at most 2. We provide plenty of examples that show that several features of the symplectic Grassmannian do not hold after tropicalizing. We show exactly when do conormal fans of matroids satisfy these characterizations, as well as doing the same for a non-constant coefficient generalization. Finally, we propose several directions to extend the study of the tropical symplectic Grassmannian.

math.CO↗