Perverse schobers are categorifications of perverse sheaves. We construct a perverse schober on a partial compactification of the stringy Kähler moduli space (SKMS) associated by Halpern-Leistner and Sam to a quasi-symmetric representation X of a reductive group G, extending the local system of triangulated categories exhibited by them. The triangulated categories appearing in our perverse schober are subcategories of the derived category of the quotient stack X/G.

We give a more conceptual construction of a comparison algebra morphism from the K-theoretical Hall algebra to a twist of the cohomological Hall algebra associated to a symmetric quiver, and extend the result to quivers with potential.

We find an explicit S-n -equivariant bijection between the integral points in a certain zonotope in R- n , combinatorially equivalent to the permutahedron, and the set of m-parking functions of length n. This bijection restricts to a bijection between the regular S n -orbits and (m, n)-Dyck paths, the number of which is given by the Fuss-Catalan number A( n) (m, 1). Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes.

We give a brief review of the cohomological Hall algebra CoHA H and the K-theoretical Hall algebra KHA R associated to quivers. In the case of symmetric quivers, we show that there exists a homomorphism of algebras (obtained from a Chern character map) R→Hˆσ˜ where Hˆσ˜ is a Zhang twist of the completion of H. Moreover, we establish the equivalence of categories of “locally finite” graded modules H-Modlf≃RQ-Modlf. Examples of locally finite Hˆ-, resp. RQ-modules appear naturally as the cohomology, resp. K-theory, of framed moduli spaces of quivers.

Let X be a projective crepant resolution of a Gorenstein affine toric variety and let ((ℂ^*)^k,f) be the LG-model which is the Hori-Vafa mirror dual of X. Let D be a generic fiber of f equipped with the restriction of the standard Liouville form on (ℂ^*)^k. Let 𝒦_A be the so-called "stringy Kähler moduli space" of X. We show that π_1(𝒦_A) acts on the wrapped Fukaya category of D. Using results by Gammage - Shende and Zhou, this result implies that π_1(𝒦_A) acts on D^b(coh(∂ X)) where ∂ X is the toric boundary divisor of X. We show that the induced action of π_1(𝒦_A) on K_0(coh(∂ X)) may be extended in a natural way to an action on K_0(X) which corresponds to a GKZ system.

We show that for a gradable finite dimensional algebra the perfect complexes and bounded derived category cannot be distinguished by homotopy invariants.

The derived category of an algebraic variety might be a source of a myriad of new (categorical) symmetries. Some are predicted by homological mirror symmetry, to be obtained from the fundamental group of the space of complex structures of its mirror partner. These finally lead to differential equations. We expositorily unravel a part of this conjectural master plan for a class of toric varieties.

Let a reductive group G act on a smooth variety X such that a good quotient X//G exists. We show that the derived category of a noncommutative crepant resolution (NCCR) of X//G, obtained from a G-equivariant vector bundle on X, can be embedded in the derived category of the (canonical, stacky) Kirwan resolution of X//G. In fact, the embedding can be completed to a semi-orthogonal decomposition in which the other parts are all derived categories of Azumaya algebras over smooth Deligne-Mumford stacks.

We find an explicit S_n-equivariant bijection between the integral points in a certain zonotope in ℝ^n, combinatorially equivalent to the permutahedron, and the set of m-parking functions of length n. This bijection restricts to a bijection between the regular S_n-orbits and (m,n)-Dyck paths, the number of which is given by the Fuss-Catalan number A_n(m,1). Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes.

Let $A$ be an integral matrix and let $P$ be the convex hull of its columns. By a result of Gelfand, Kapranov and Zelevinski, the so-called principal $A$-determinant locus is equal to the union of the closures of the discriminant loci of the Laurent polynomials associated to the faces of $P$ that are hypersurfaces. In this short note we show that it is also the straightforward union of all the discriminant loci, i.e. we may include those of higher codimension, and there is no need to take closures. This answers a question by Kite and Segal.

Perverse schobers are categorifications of perverse sheaves. In prior work we constructed a perverse schober on a partial compactification of the stringy Kähler moduli space (SKMS) associated by Halpern-Leistner and Sam to a quasi-symmetric representation of a reductive group. When the group is a torus the SKMS corresponds to the complement of the GKZ discriminant locus (which is a hyperplane arrangement in the quasi-symmetric case shown by Kite). We show here that a suitable variation of the perverse schober we constructed provides a categorification of the associated GKZ hypergeometric system in the case of non-resonant parameters. As an intermediate result we give a description of the monodromy of such "quasi-symmetric" GKZ hypergeometric systems.

Let $R$ be the homogeneous coordinate ring of the Grassmannian $\mathbb{G}=Gr(2,n)$ defined over an algebraically closed field $k$ of characteristic $p \geq \max\{n-2,3\}$. In this paper we give a description of the decomposition of $R$, considered as graded $R^{p^r}$-module, for $r \geq 2$. This is a companion paper to our earlier paper, where the case $r=1$ was treated, and taken together, our results imply that $R$ has finite F-representation type (FFRT). Though it is expected that all rings of invariants for reductive groups have FFRT, ours is the first non-trivial example of such a ring for a group which is not linearly reductive. As a corollary, we show that the ring of differential operators $D_k(R)$ is simple, that $\mathbb{G}$ has global finite F-representation type (GFFRT) and that $R$ provides a noncommutative resolution for $R^{p^r}$.

We show that all toric noncommutative crepant resolutions (NCCRs) of affine GIT quotients of "weakly symmetric" unimodular torus representations are derived equivalent. This yields evidence for a non-commutative extension of a well known conjecture by Bondal and Orlov stating that all crepant resolutions of a Gorenstein singularity are derived equivalent. We prove our result by showing that all toric NCCRs of the affine GIT quotient are derived equivalent to a fixed Deligne-Mumford GIT quotient stack associated to a generic character of the torus. This extends a result by Halpern-Leistner and Sam which showed that such GIT quotient stacks are a geometric incarnation of a family of specific toric NCCRs constructed earlier by the authors.

Recently McBreen and Webster constructed a tilting bundle on a smooth hypertoric variety (using reduction to finite characteristic) and showed that its endomorphism ring is Koszul. In this short note we present alternative proofs for these results. We simply observe that the tilting bundle constructed by Halpern-Leistner and Sam on a generic open Geometric Invariant Theory substack of the ambient linear space restricts to a tilting bundle on the hypertoric variety. The fact that the hypertoric variety is defined by a quadratic regular sequence then also yields an easy proof of Koszulity.

If G is a reductive group which acts on a linearized smooth scheme $X$ then we show that under suitable standard conditions the derived category of coherent sheaves of the corresponding GIT quotient stack $X^{ss}/G$ has a semi-orthogonal decomposition consisting of derived categories of coherent sheaves of rings on the categorical quotient $X^{ss}/\!/G$ which are locally of finite global dimension. One of the components of the decomposition is a certain non-commutative resolution of $X^{ss}/\!/G$ constructed earlier by the authors. As a concrete example we obtain in the case of odd Pfaffians a semi-orthogonal decomposition of the corresponding quotient stack in which all the parts are certain specific non-commutative crepant resolutions of Pfaffians of lower or equal rank which had also been constructed earlier by the authors. In particular this semi-orthogonal decomposition cannot be refined further since its parts are Calabi-Yau. The results in this paper also complement a result by Halpern-Leistner (and similar results by Ballard-Favero-Katzarkov and Donovan-Segal) that asserts the existence of a semi-orthogonal decomposition of the derived category of $X/G$ in which one of the components is the derived category of $X^{ss}/G$.

Using the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein affine toric variety Spec R admits a toric non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of Spec R using standard toric methods. Our proof does not use dimer models.

In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra ([5, 6]). In particular, in the case of the Lie algebra sl2(C), there is a correspondence between Verma modules and certain line modules that associates a pair (h, φ), where h is a two-dimensional Lie subalgebra of sl2(C) and φ ∈ h∗ satisfies φ([h, h]) = 0, to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra sl(1|1) and for a color Lie algebra associated to the Lie algebra sl2.

In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) , there is a correspondence between Verma modules and certain line modules that associates a pair ( h , ϕ ) (\mathfrak {h},\,\phi ) , where h \mathfrak {h} is a 2 2 -dimensional Lie subalgebra of s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) and ϕ ∈ h ∗ \phi \in \mathfrak {h}^* satisfies ϕ ( [ h , h ] ) = 0 \phi ([\mathfrak {h}, \, \mathfrak {h}]) = 0 , to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra s l ( 1 | 1 ) \mathfrak {sl}(1|1) and for a color Lie algebra associated to the Lie algebra s l 2 \mathfrak {sl}_2 .

In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra sl(2)(C), there is a correspondence between Verma modules and certain line modules that associates a pair (h, phi), where h is a 2-dimensional Lie subalgebra of sl(2)(C) and phi is an element of h* satisfies phi([h, h]) = 0, to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra sl(1 vertical bar 1) and for a color Lie algebra associated to the Lie algebra sl(2).

Positivstellensatze are fundamental results in real algebraic geometry providing algebraic certificates for positivity of polynomials on semialgebraic sets. In this article, Positivstellensatze for trace polynomials positive on semialgebraic sets of nxn matrices are provided. A Krivine-Stengle-type Positivstellensatz is proved characterizing trace polynomials nonnegative on a general semialgebraic set K using weighted sums of Hermitian squares with denominators. The weights in these certificates are obtained from generators of K and traces of Hermitian squares. For compact semialgebraic sets K Schmudgen- and Putinar-type Positivstellensatze are obtained: every trace polynomial positive on K has a sum of Hermitian squares decomposition with weights and without denominators. The methods employed are inspired by invariant theory, classical real algebraic geometry and functional analysis. Procesi and Schacher in 1976 developed a theory of orderings and positivity on central simple algebras with involution and posed a Hilbert's 17th problem for a universal central simple algebra of degree n: is every totally positive element a sum of Hermitian squares? They gave an affirmative answer for n=2. In this paper, a negative answer for n=3 is presented. Consequently, including traces of Hermitian squares as weights in the Positivstellensatze is indispensable.