Article
Mathematics
Ryan Kinser, Jenna Rajchgot
Summary: This paper unifies the equivariant geometry of type D quiver representation varieties, double Grassmannians, and symmetric varieties GL(a + b)/GL(a) x GL(b), by translating results about singularities of orbit closures, combinatorics of orbit closure containment, and torus equivariant K-theory between these three families. These results are all obtained from a generalization of Zelevinsky's construction for type A quivers to the type D setting, by giving explicit embeddings with nice properties of homogeneous fiber bundles over type D quiver representation varieties into these symmetric varieties.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics
Mee Seong Im, Chun-Ju Lai, Arik Wilbert
Summary: The Maffei-Nakajima theorem demonstrates the realization of the Slodowy variety as a Nakajima quiver variety of type A, showing an implicit isomorphism that involves solving a system of equations with linear maps variables. This paper constructs solutions to this system and provides an explicit and efficient way to compute the image of a complete flag within the Slodowy variety under the Maffei-Nakajima isomorphism, while describing these flags in terms of quiver representations.
JOURNAL OF ALGEBRA
(2022)
Article
Mathematics
Giovanni Cerulli Irelli, Francesco Esposito, Hans Franzen, Markus Reineke
Summary: The article demonstrates properties of quiver Grassmannians associated with rigid quiver representations, including properties of the cohomology ring, Chow ring generators, and polynomial point count. By restricting to finite or affine type quivers, it is shown that quiver Grassmannians associated with indecomposable representations have a cellular decomposition. Additionally, the geometry behind the cluster multiplication formula of Caldero and Keller is studied, providing a new proof of a slightly more general result.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics
Evgeny Feigin, Martina Lanini, Alexander Puetz
Summary: In this paper, we study the properties of certain quiver Grassmannians for the cyclic quiver, including their cellular decomposition, description of irreducible components, and construction of resolutions for singularities.
CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES
(2023)
Article
Mathematics
Alexander Blose, Patricia Klein, Owen Mcgrath, A. N. D. Jackson Morris
Summary: We examine Li's double determinantal varieties in the special case that they are toric, showing that they are irreducible and smooth. We provide a straightforward formula for their dimension and offer empirical evidence concerning an open problem in local algebra using the smallest nontrivial toric double determinantal variety.
COMMUNICATIONS IN ALGEBRA
(2021)
Article
Mathematics
Hunter Dinkins, Andrey Smirnov
Summary: In this paper, we investigate the capped vertex functions associated with certain zero-dimensional type -A Nakajima quiver varieties. We derive explicit combinatorial formulas for the capped vertex functions by inserting descendants using the Macdonald operators. We determine the monodromy of the vertex functions and establish its coincidence with the elliptic R-matrix of the symplectic dual variety. We also apply our findings to compute the vertex functions and characters of tautological bundles on quiver varieties formed from arbitrary stability conditions.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics
Hans Franzen
Summary: This paper demonstrates that any quiver Grassmannian associated with a rigid representation of a quiver is a rational variety, employing torus localization techniques.
ALGEBRAS AND REPRESENTATION THEORY
(2022)
Article
Mathematics
Alexander Kuznetsov, Maxim Smirnov
Summary: In this study, we extend and provide support for a conjecture regarding the relationship between the small quantum cohomology ring and derived category of coherent sheaves of a smooth Fano variety. By examining specific examples, we demonstrate the validity of this conjecture in the context of (co)adjoint homogeneous varieties of simple algebraic groups.
COMPOSITIO MATHEMATICA
(2021)
Article
Mathematics
Hunter Dinkins, Andrey Smirnov
Summary: This article discusses the moduli spaces of quasimaps to zero-dimensional A(infinity) Nakajima quiver varieties and obtains an explicit combinatorial formula for the equivariant Euler characteristic of these moduli spaces. Applications to symplectic duality are also discussed.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2022)
Article
Mathematics, Applied
Gwyn Bellamy, Travis Schedler
Summary: In this article, Nakajima quiver varieties are considered from the perspective of symplectic algebraic geometry. The study proves that they are all symplectic singularities according to Beauville and completely classifies which ones admit symplectic resolutions. Additionally, it is shown that the smooth locus aligns with the locus of canonically theta-polystable points, expanding on a previous result by Le Bruyn. The study also examines the etale local structure of these varieties and identifies their symplectic leaves, with an interesting finding that not all symplectic resolutions of quiver varieties seem to originate from variation of GIT.
SELECTA MATHEMATICA-NEW SERIES
(2021)
Article
Mathematics
Ivan Mirkovic, Maxim Vybornov, Vasily Krylov
Summary: In type A, we identify the equivalences of geometries in three settings: Nakajima's (framed) quiver varieties, conjugacy classes of matrices, and loop Grassmannians. These equivalences are expressed through explicit formulas. In particular, we embed the framed quiver varieties into Beilinson-Drinfeld Grassmannians, leading to a compactification of Nakajima varieties and a decomposition of affine Grassmannians into Nakajima varieties. As an application, we present a geometric interpretation of symmetric and skew (GL(m), GL(n)) dualities.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Dalton Bidleman, Luke Oeding
Summary: Restricted secant varieties of Grassmannians are constructed by summing points corresponding to k-planes with the restriction of a prescribed intersection dimension. We study the dimensions of these restricted secant varieties and relate them to the dimensions of secants of Grassmannians using an incidence variety construction. We introduce the concept of expected dimension and provide a formula for the dimension of all restricted secant varieties of Grassmannians, assuming the non-defectivity conjecture on Grassmannians holds. We demonstrate example calculations in Macaulay 2 and suggest ways to improve computational efficiency. We also discuss a potential application to coding theory.
COLLECTANEA MATHEMATICA
(2023)
Article
Mathematics
Jun-Muk Hwang, Qifeng Li
Summary: In this paper, we prove that the variety of minimal rational tangents (VMRT) at a general point of a uniruled projective manifold is projectively equivalent to a symplectic or an odd-symplectic Grassmannian. We show that a general minimal rational curve is biholomorphic to a general line in a presymplectic Grassmannian. Furthermore, we use a vanishing condition and the geometry of minimal rational curves to extend Tanaka's method to characterize symplectic and odd-symplectic Grassmannians beyond parabolic geometries.
JOURNAL OF DIFFERENTIAL GEOMETRY
(2021)
Article
Mathematics
Katsuyuki Naoi
Summary: The generalized quantum affine Schur-Weyl duality functor establishes an equivalence between two different finite-dimensional module categories, which is of significant importance in the field of algebra.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics, Applied
Rasool Hafezi, Yi Zhang
Summary: In this paper, we investigate the components of the stable Auslander-Reiten quiver of a certain subcategory of monomorphism category S(Gprj-Lambda) that contains boundary vertices. We describe the shape of these components and demonstrate that certain components are connected to the orbits of an auto-equivalence on the stable category Gprj-Lambda. Moreover, we show that under certain conditions, the cardinalities of finite components are divisible by 3, suggesting a recurring three-periodicity phenomenon.
SCIENCE CHINA-MATHEMATICS
(2023)
Article
Mathematics, Applied
Sarah Scherotzke, Nicolo Sibilla
SELECTA MATHEMATICA-NEW SERIES
(2016)
Article
Mathematics
Bernhard Keller, Sarah Scherotzke
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
(2016)
Article
Mathematics
Sarah Scherotzke, Nicolo Sibilla
PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY
(2016)
Article
Mathematics
Sarah Scherotzke
COLLOQUIUM MATHEMATICUM
(2016)
Article
Mathematics
David Carchedi, Sarah Scherotzke, Nicolo Sibilla, Mattia Talpo
GEOMETRY & TOPOLOGY
(2017)
Article
Mathematics
David Carchedi, Sarah Scherotzke, Nicolo Sibilla, Mattia Talpo
GEOMETRY & TOPOLOGY
(2017)
Article
Mathematics
Sarah Scherotzke, Nicolo Sibilla, Mattia Talpo
COMPOSITIO MATHEMATICA
(2018)
Article
Mathematics
Sarah Scherotzke
MATHEMATISCHE ZEITSCHRIFT
(2019)
Article
Mathematics, Applied
Sarah Scherotzke, Nicolo Sibilla
JOURNAL OF NONCOMMUTATIVE GEOMETRY
(2019)
Article
Mathematics
Sarah Scherotzke, Nicolo Sibilla, Mattia Talpo
JOURNAL OF ALGEBRA
(2020)
Article
Mathematics
Sarah Scherotzke, Peter Schneider
Summary: This article studies the derived version of the classical parabolic induction functor, and proves that the derived category of smooth G-G representations is equivalent to a certain differential graded k-algebra. It constructs a derived parabolic induction functor on the dg Hecke algebra side and discusses its adjoint functors.
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY
(2022)
Article
Mathematics, Applied
Tristan Bozec, Damien Calaque, Sarah Scherotzke
Summary: This study demonstrates the connection between relative Calabi-Yau structures on noncommutative moment maps and (quasi-)bisymplectic structures. The authors also apply this connection to investigate the Poisson structures on the moduli spaces of representations of deformed multiplicative preprojective algebras.
FORUM OF MATHEMATICS SIGMA
(2023)
Article
Mathematics
Marc Hoyois, Pavel Safronov, Sarah Scherotzke, Nicolo Sibilla
Summary: The paper proves a categorification of the Grothendieck-Riemann-Roch theorem, establishing a connection with Toen and Vezzosi's secondary Chern character. It compares the Toen-Vezzosi Chern character with the classical Chern character, demonstrating that the categorified Chern character can recover the classical de Rham realization.
COMPOSITIO MATHEMATICA
(2021)
Article
Mathematics
Marc Hoyois, Sarah Scherotzke, Nicolo Sibilla
ADVANCES IN MATHEMATICS
(2017)
