Modelling classical types: Algebraic group actions via algebras with symmetries
Final Report Abstract
Our project is situated in the field of algebraic foundational research, at the intersection of representation theory of associative algebras (via quivers with relations) with Lie theory and algebraic geometry. For a long time, the fact that the theory of quiver representations is defined in type A (i.e., for general linear groups and their Lie algebras) could be seen as a limitation. To overcome this restriction, Derksen and Weyman introduced the concept of symmetric quivers in 2002. This approach allows us to consider classical analogies, particularly symplectic or orthogonal representations, which are also referred to as symmetric quiver representations. These representations are collected in so-called symmetric representation varieties on which reductive groups act by base change. Our main focus in this project lies on the orbits of the aforementioned group actions, on their closures, and on applications of the theory. The orbits correspond to the isomorphism classes of (symmetric) representations. It is known from 1 that the symmetric orbits are induced by restriction, and we prove this fact using new methods. Regarding orbit closures, it remains open under which assumptions they are induced. We define the associative representation-finite see-saw algebra and provide an explicit example of two symmetric representations whose partial closure ordering is not induced in the orthogonal case. Subsequently, we introduce the symmetric Ext-order, which serves as a sufficient criterion for symmetric closure relations. A necessary criterion for symmetric closure relations is the Hom-order, already known from the non-symmetric context. One of our main results states that for a symmetric, representation-finite quiver (without relations), all these partial orders are equivalent. A symmetric representation is included in the closure of another with respect to the symmetric representation variety if and only if it is also included with respect to the entire representation variety. From this, we deduce that the symmetric closure order is equivalent to the symmetric Ext-order and equivalent to the Hom-order. In the case of an equioriented Dynkin type A quiver, we concretely prove the theorem using combinatorial methods such as cuts and shifts. Subsequently, we focus on applications and conceptualize linear degenerate symplectic flag varieties as symmetric degenerations within the framework of equioriented type A quivers. Furthermore, we demonstrate that a PBW degeneration of the symplectic flag variety is irreducible, reduced, normal, Cohen-Macaulay, and Frobenius-split, and it also exhibits rational singularities.
Publications
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Approaching symplectic/orthogonal orbit closure relations. Oberwolfach Reports, No. 23/2022, Mathematisches Forschungsinstitut Oberwolfach
Boos, Magdalena
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Symmetric degenerations are not in general induced by type A degenerations. Rendiconti di Matematica e delle sue Applicazioni (7). Volume 43, 2022
Boos, Magdalena & Cerulli, Irelli Giovanni
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Linear degenerate symplectic flag varieties: symmetric degenerations and PBW locus
Boos, Magdalena; Cerulli, Irelli Giovanni; Fang, Xin & Fourier, Ghislain
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On degenerations and extensions of symplectic and orthogonal quiver representations. Arkiv för Matematik, 63(1), 61-115.
Boos, Magdalena & Cerulli, Irelli Giovanni
