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TY - THES AU - Iezzi, Giulia TI - Quiver Grassmannians: Schubert varieties, linear degenerations and tropical geometry PB - RWTH Aachen University VL - Dissertation CY - Aachen M1 - RWTH-2025-08020 SP - 1 Online-Ressource : Illustrationen PY - 2025 N1 - Cotutelle-Dissertation. - Veröffentlicht auf dem Publikationsserver der RWTH Aachen University N1 - Dissertation, RWTH Aachen University, 2025. - Dissertation, Universität Tor Vergata, 2025 AB - Quivers and their representations theory provide powerful tools, in particular for studying representations of finite-dimensional algebras; they were introduced to treat problems of linear algebra, but present rich connections to diverse mathematical subjects. A key advantage of using quivers for studying problems of geometrical nature is the possibility to exploit combinatorial and algebraic tools to deduce geometric properties of the associated projective variety, the quiver Grassmannian. The main focus of the research presented in this thesis lies at the intersection of algebra, geometry and representation theory. The aim of this thesis is, firstly, to find a realisation of Schubert varieties as quiver Grassmannians by means of a quiver and quiver representation with certain reasonable properties. Subsequently, this realisation is exploited to def ine linear degenerations of Schubert varieties. Furthermore, we generalise the construction of the flag Dressian by defining the concept of quiver Dressian and compare it to the tropicalisation of the corresponding quiver Grassmannian. Chapter 1 covers the necessary background for the study of quivers and quiver representations, both from a categorical and a from a geometric point of view. In Chapter 2, we consider a special quiver with relations and construct a rigid representation of this quiver. We study a certain subvariety of the variety of representations, describing the decompositions into indecomposables for the elements of this subvariety and parametrising the B-isomorphism classes, where B represents the Borel subgroup of upper-triangular matrices. Chapter 3 summarises basic facts and definitions about quiver Grassmannians, flag varieties and their Schubert varieties. We prove that any quiver Grassmannian associated to the quiver representation defined in Chapter 2 is smooth and irreducible, and its dimension can be easily computed by means of the Euler-Ringel form. Chapter 4 contains several of our main results. We prove that every permutation admits a geometrically compatible decomposition- we introduce this notion in Section 4.1- and realise the Bott-Samelson resolution of a fixed Schubert variety using the quiver representation and quiver Grassmannian considered previously. Lastly, by choosing a different, appropriate dimension vector for our quiver, we give an explicit isomorphism between any chosen smooth Schubert variety and the corresponding quiver Grassmannian. In Chapter 5, we explore linear degenerations. The first section briefly recalls linear degenerations of flag varieties, while in the second section we build upon the constructions and results obtained in Chapter 2, 3 and 4 and define linear degenerations of Schubert varieties. We show how one of the parametrisations considered in Section 2.3 describes the relations between the B-orbits (and their closures) in the subvariety defined in Section 2.2 and list the conditions that are necessary and sufficient for a tuple of non-negative integers to be the parametrisation of some representation in this subvariety. Finally, we open the discussion on the flat locus of the projection from the universal linear degeneration onto the considered representation space. We present and motivate a conjecture about this flat locus. Chapter 6 is dedicated to a bridge between quiver representation theory and tropical geometry, in particular to the introduction of quivers of valuated matroids and the study of their tropical parameter spaces. We define quiver Dressians, which parametrise containment of tropical linear spaces after tropical matrix multiplication, and show that tropicalisations of quiver Grassmannians parametrise the realisable analogue. We further introduce affine morphisms of valuated matroids and show compatibility with weakly monomial quiver representations. Finally, we show that, starting in ambient dimension 2, quiver Dressians can have nonrealisable points. LB - PUB:(DE-HGF)11 DO - DOI:10.18154/RWTH-2025-08020 UR - https://publications.rwth-aachen.de/record/1018850 ER -