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Giovanni Cerulli Irelli

Publications and source records attributed to Giovanni Cerulli Irelli.

At least 19 recordsLinked to original sources
arXiv2026

Extremality of principal quiver Grassmannians

We study families of quiver Grassmannians, parametrizing subrepresentations of representations of Dyn\-kin quivers, which are naturally associated to a projective and an injective representation. Our main result is an explicit description of the flat irreducible locus of such a family. More precisely, we prove that a member of this family is irreducible and of the expected dimension if and only if the ambient quiver representation degenerates to the direct sum of the defining projective and injective representations.

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arXiv2026

A categorification of Kauffman states for planar graphs

Given a decorated planar graph $(G,ω)$, where $G$ is a planar graph and $ω\in H^1(|\mathcal{Q}G|,\mathbb{Z})$ with $\mathcal{Q}G$ the directed medial graph of $G$, we call some angular functions $ω$-compatible and study two distinct but related directed graphs: $\mathcal{L}(G,ω)$, which is the directed graph of such functions, and $BMS(G,ω)$, the directed graph of BMS states which are some pairs of $ω$-compatible functions plus additional data. We give sufficient conditions for $\mathcal{L}(G,ω)$ to be a graded distributive lattice, recovering Kauffman's Clock Theorem when $G$ is a knot diagram. We also define a potential on $\mathcal{Q} G$ and associate a representation of the corresponding quiver with potential to every BMS state. Under suitable assumptions, this construction yields an isomorphism between $\mathcal{L}(G,ω)$ and the lattice of subrepresentations of a maximal representation, generalizing a result of Bazier-Matte--Schiffler.

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arXiv2026

Polynomial identities for quivers via incidence algebras

We show that the path algebra of a quiver satisfies the same polynomial identities of an algebra of matrices, if any. In particular, the algebra of nxn matrices is PI-equivalent to the path algebra of the oriented cycle with n vertices.

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arXiv2025

Quivers with Polynomial Identities

We provide a topological characterization of quivers whose path algebra satisfies a polynomial identity. This class includes the oriented cycle and acyclic quivers and, in the latter case, we describe the associated T-ideal. We introduce a generalization of Arnold's A-graded algebras, which we call locally A-graded algebras, and prove that they are also PI. We give an example of a quiver algebra satisfying a polynomial identity, even if the path algebra of the quiver does not.

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arXiv2024

Specialization map for quiver Grassmannians

We define a specialization map between cohomology algebras of quiver Grassmannians of Dynkin type and we prove that it is surjective in type A. This generalizes a result of Lanini and Strickland.

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arXiv2024

Linear degenerate symplectic flag varieties: symmetric degenerations and PBW locus

We conceptualize in the paper the linear degenerate symplectic flag varieties as symmetric degenerations within the framework of type $A$ equioriented quivers. First, in the larger context of symmetric degenerations, we give a self-contained proof of the equivalence of different degeneration orders. Furthermore, we investigate the PBW locus: geometric properties of the degenerate varieties in this locus are proved by realizing them from different perspectives.

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arXiv2022

Symmetric degenerations are not in general induced by type A degenerations

We consider a symmetric quiver with relations. Its (symmetric) representations of a fixed symmetric dimension vector are encoded in the (symmetric) representation varieties. The orbits by a (symmetric) base change group action are the isomorphism classes of (symmetric) representations. The symmetric orbits are induced by simply restricting the nonsymmetric orbits. However, when it comes to orbit closure relations, it is so far an open question under which assumptions they are induced. In connection with Borel orbits of 2-nilpotent matrices of classical Lie algebras, we describe an explicit example of a quiver of finite representation type for which orbit closure relations are induced in types B and C, but not in type D.

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arXiv2021

Motzkin combinatorics in linear degenerations of the flag variety

We establish an explicit combinatorial/homological characterization of supports for linear degenerations of flag varieties. For such purpose, we introduce the concept of an excessive multisegment. It provides a new class of combinatorial objects counted by Motzkin numbers.

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arXiv2021

On degenerations and extensions of symplectic and orthogonal quiver representations

We discuss degenerations of symplectic and orthogonal representations of symmetric quivers and algebras with self-dualities. As in the non-symmetric case, we define a partial ordering, that we call symmetric Ext-order which gives a sufficient criterion for a symmetric degeneration. Then a detailed discussion of type A quivers and their (symmetric) representation theory via Auslander-Reiten theory leads to our main theorem which states that the symmetric degeneration order of a symmetric quiver of finite type is induced by the usual degeneration order between representations of the underlying quiver.

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arXiv2020

Three lectures on quiver Grassmannians

This paper contains the material discussed in the series of three lectures that I gave during the workshop of the ICRA 2018 in Prague. I will introduce the reader to some of the techniques used in the study of the geometry of quiver Grassmannians. The notes are quite elementary and thought for phd students or young researchers. I assume that the reader is familiar with the representation theory of quivers.

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arXiv2019

Cell decompositions and algebraicity of cohomology for quiver Grassmannians

We show that the cohomology ring of a quiver Grassmannian asssociated with a rigid quiver representation has property (S): there is no odd cohomology and the cycle map is an isomorphism; moreover, its Chow ring admits explicit generators defined over any field. From this we deduce the polynomial point count property. By restricting the quiver to finite or affine type, we are able to show a much stronger assertion: namely, that a quiver Grassmannian associated to an indecomposable (not necessarily rigid) representation admits a cellular decomposition. As a corollary, we establish a cellular decomposition for quiver Grassmannians associated with representations with rigid regular part. Finally, we study the geometry behind the cluster multiplication formula of Caldero and Keller, providing a new proof of a slightly more general result.

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arXiv2019

Parabolic orbits of $2$-nilpotent elements for classical groups

We consider the conjugation-action of the Borel subgroup of the symplectic or the orthogonal group on the variety of nilpotent complex elements of nilpotency degree $2$ in its Lie algebra. We translate the setup to a representation-theoretic context in the language of a symmetric quiver algebra. This makes it possible to provide a parametrization of the orbits via a combinatorial tool that we call symplectic/orthogonal oriented link patterns. We deduce information about numerology. We then generalize these classifications to standard parabolic subgroups for all classical groups. Finally, our results are restricted to the nilradical.

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arXiv2019

Linear degenerations of flag varieties: partial flags, defining equations, and group actions

We continue, generalize and expand our study of linear degenerations of flag varieties from [G. Cerulli Irelli, X. Fang, E. Feigin, G. Fourier, M. Reineke, Math. Z. 287 (2017), no. 1-2, 615-654]. We realize partial flag varieties as quiver Grassmannians for equi-oriented type A quivers and construct linear degenerations by varying the corresponding quiver representation. We prove that there exists the deepest flat degeneration and the deepest flat irreducible degeneration: the former is the partial analogue of the mf-degenerate flag variety and the latter coincides with the partial PBW-degenerate flag variety. We compute the generating function of the number of orbits in the flat irreducible locus and study the natural family of line bundles on the degenerations from the flat irreducible locus. We also describe explicitly the reduced scheme structure on these degenerations and conjecture that similar results hold for the whole flat locus. Finally, we prove an analogue of the Borel-Weil theorem for the flat irreducible locus.

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arXiv2016

Linear degenerations of flag varieties

Linear degenerate flag varieties are degenerations of flag varieties as quiver Grassmannians. For type A flag varieties, we obtain characterizations of flatness, irreducibility and normality of these degenerations via rank tuples. Some of them are shown to be isomorphic to Schubert varieties and can be realized as highest weight orbits of partially degenerate Lie algebras, generalizing the corresponding results on degenerate flag varieties. To study normality, cell decompositions of quiver Grassmannians are constructed in a wider context of equioriented quivers of type A.

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arXiv2016

Geometry of Quiver Grassmannians of Dynkin type with applications to cluster algebras

The paper includes a new proof of the fact that quiver Grassmannians associated with rigid representations of Dynkin quivers do not have cohomology in odd degrees. Moreover, it is shown that they do not have torsion in homology. A new proof of the Caldero--Chapoton formula is provided. As a consequence a new proof of the positivity of cluster monomials in the acyclic clusters associated with Dynkin quivers is obtained. The methods used here are based on joint works with Markus Reineke and Evgeny Feigin.

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arXiv2016

Degenerate flag varieties and Schubert varieties: a characteristic free approach

We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for group schemes of the same type and doubled rank. We deduce that the corresponding degenerate flag varieties are isomorphic to Schubert varieties in any characteristic.

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arXiv2015

Schubert Quiver Grassmannians

Quiver Grassmannians are projective varieties parametrizing subrepresentations of given dimension in a quiver representation. We define a class of quiver Grassmannians generalizing those which realize degenerate flag varieties. We show that each irreducible component of the quiver Grassmannians in question is isomorphic to a Schubert variety. We give an explicit description of the set of irreducible components, identify all the Schubert varieties arising, and compute the Poincaré polynomials of these quiver Grassmannians.

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