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Markus Reineke

Publications and source records attributed to Markus Reineke.

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.

math.RT↗
arXiv2026

Fano 4-fold quiver moduli from subspace quivers

We classify the moduli spaces of representations of subspace quivers which are Fano fourfolds, under a natural assumption on the dimension vector. These moduli spaces can also be described as GIT quotients of products of Grassmannians by the diagonal action of a projective linear group, and there are exactly four of them. They are rational, of pure Hodge-Tate type, infinitesimally rigid, and have finite automorphism groups, with Picard ranks 5, 6, 6 and 7, making them interesting examples in the classification of Fano fourfolds of large Picard rank, as they are not toric or products. Two are known varieties: Manivel's Segre cousin of the Segre cubic 3-fold, and the Fano model of the blowup of $\mathbb{P}^4$ in six points. The other two appear to be new: one is an involution surface bundle over $\mathbb{P}^2$, and the other is a "Segre cousin once-removed", whose geometry closely parallels that of the Segre cousin. Using techniques from quiver moduli, which we survey, we describe the geometry of all four fourfolds in detail.

math.AG↗
arXiv2026

Two remarks on decomposition numbers of standard modules for quantum affine $\mathfrak{sl}_2$

We use Nakajima's geometric approach to representations of quantum affine algebras and recent results on explicit descriptions of specific canonical basis elements, to derive closed positive formulas for certain decomposition numbers of representations of quantum affine $\mathfrak{sl}_2$. Moreover, we obtain a piecewise-linear closed formula for the $q$-characters of irreducible representations of quantum affine $\mathfrak{sl}_2$.

math.QA↗
arXiv2025

Cluster scattering diagrams via quiver moduli and tight gradings

We study rank-2 cluster scattering diagrams through moduli spaces of quiver representations and a recently developed combinatorial framework of tight gradings. Combining quiver-theoretic and combinatorial methods, we prove and extend a collection of conjectures posed by Elgin--Reading--Stella concerning the structural and enumerative properties of the wall-function coefficients. The tight grading perspective also provides a new proof of the Weyl group symmetry of the scattering diagram.

math.CO↗
arXiv2025

Punctual noncommutative Hilbert schemes

Punctual noncommutative Hilbert schemes are projective varieties parametrizing finite codimensional left ideals in noncommutative formal power series rings. We determine their motives and intersection cohomology, by constructing affine pavings and small resolutions of singularities.

math.AG↗
arXiv2025

From brick manifolds to Grassmannians of bimodules

We study a class of Grassmannians of sub-bimodules over the path algebras of quivers. Our quiver Grassmannians include Escobar's brick manifolds as well as Labelle's generalizations. We give an explicit construction of the varieties in question, provide examples and clarify connection with the quiver representation spaces. We also prove smoothness of our Grassmannians, construct cellular decompositions and derive a realization as framed moduli spaces. The framed moduli realization leads to a recursive formula for the motives.

math.RT↗
arXiv2025

Counting generating spaces of matrices

We prove that generating subspaces of matrix rings over finite fields are counted by polynomials. We use this result to define and study two-variable versions of polynomials counting isomorphism classes of absolutely irreducible representations of free algebras.

math.RT↗
arXiv2025

A-D-E diagrams, Hodge--Tate hyperplane sections and semisimple quantum cohomology

It is known that the semisimplicity of quantum cohomology implies the vanishing of off-diagonal Hodge numbers (Hodge--Tateness). We investigate which hyperplane sections of homogeneous varieties possess either of the two properties. We provide a new efficient criterion for non-semisimplicity of the small quantum cohomology ring of Fano manifolds that depends only on the Fano index and Betti numbers. We construct a bijection between Dynkin diagrams of types A, D or E, and complex Grassmannians with Hodge-Tate smooth hyperplane sections. By applying our criteria and using monodromy action, we completely characterize the semisimplicity of the small quantum cohomology of smooth hyperplane sections in the case of complex Grassmannians, and verify a conjecture of Benedetti and Perrin in the case of (co)adjoint Grassmannians.

math.AG↗
arXiv2025

Local intersection cohomology of varieties of complexes

We compute the local intersection cohomology of the irreducible components of varieties of complexes, by using Lusztig's geometric approach to quantum groups and explicit constructions of elements of Lusztig's canonical bases.

math.AG↗
arXiv2024

Expander representations of quivers

We propose a definition of expander representations of quivers, generalizing dimension (or linear algebra) expanders, as a qualitative refinement of slope stability. We prove existence of uniform expander representations for any wild quiver over an algebraically closed base field, using the concept of general subrepresentations and spectral properties of Cartan matrices.

math.RT↗
arXiv2024

Motives of central slope Kronecker moduli

We use dualities of quiver moduli induced by reflection functors to describe generating series of motives of Kronecker moduli spaces of central slope as solutions of algebraic and q-difference equations.

math.AG↗
arXiv2024

Categorification of quiver diagonalization and Koszul algebras

In earlier work of three of the authors of the present paper, a supercommutative quadratic algebra was associated to each symmetric quiver, and a new proof of positivity of motivic Donaldson-Thomas invariants of symmetric quivers was given using the so called numerical Koszul property of these algebras. It was furthermore conjectured that for each symmetric quiver such an algebra is Koszul. In this work, we lift the linking and unlinking operations on symmetric quivers of Ekholm, Longhi and the third author to the level of quadratic algebras, and use those lifts to prove the Koszulness conjecture.

math.RT↗
arXiv2023

Rigidity and Schofield's partial tilting conjecture for quiver moduli

We explain how Teleman quantization can be applied to moduli spaces of quiver representations to compute the higher cohomology of the endomorphism bundle of the universal bundle. We use this to prove Schofield's partial tilting conjecture, and to show that moduli spaces of quiver representations are (infinitesimally) rigid as varieties.

math.AG↗
arXiv2023

Vector fields and admissible embeddings for quiver moduli

We introduce a double framing construction for moduli spaces of quiver representations. It allows us to reduce certain sheaf cohomology computations involving the universal representation, to computations involving line bundles, making them amenable to methods from geometric invariant theory. We will use this to show that in many good situations the vector fields on the moduli space are isomorphic as a vector space to the first Hochschild cohomology of the path algebra. We also show that considering the universal representation as a Fourier-Mukai kernel in the appropriate sense gives an admissible embedding of derived categories.

math.AG↗