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Špela Špenko

Publications and source records attributed to Špela Špenko.

At least 19 recordsLinked to original sources
arXiv2026

Additive diameters of group representations

We explore the concept of additive diameters in the context of group representations, unifying various noncommutative Waring-type problems. Given a finite-dimensional representation $ρ\colon G \to \mathrm{GL}(V)$ and a subspace $U \leq V$ that generates $V$ as a $G$-module, we define the $G$-additive diameter of $V$ with respect to $U$ as the minimal number of translates of $U$ under the representation $ρ$ needed to cover $V$. We demonstrate that every irreducible representation of $\mathrm{SL}_2(\mathbf{C})$ exhibits optimal additive diameters and establish sharp bounds for the conjugation representation of $\mathrm{SL}_n(\mathbf{C})$ on its Lie algebra $\mathfrak{sl}_n(\mathbf{C})$. Additionally, we investigate analogous notions for additive diameters in Lie representations. We provide applications to additive diameters with respect to images of equivariant algebraic morphisms, linking them to the corresponding $G$-additive diameters of images of their differentials.

math.RT↗
arXiv2026

Additive diameters and covering complexity of irreducible representations

Let a group $G$ act linearly on a finite-dimensional complex vector space $V$. The group-additive diameter of a subspace $U \leq V$ is the least number of translates of $U$ whose sum is all of $V$. Counting dimensions, it is at least $\dim V / \dim U$. We show that when $G$ is compact and $V$ is irreducible, the diameter of every nonzero subspace is at most $\lceil (\dim V / \dim U) \ln \dim V \rceil$, so the trivial bound is correct up to a logarithmic factor. We measure the discrepancy by the covering complexity $\mathsf{C}(V)$, the largest ratio between the diameter of any subspace and its trivial lower bound, so that $1 \leq \mathsf{C}(V) \leq 2 + \ln \dim V$, and we determine where in this range various representations lie. Every irreducible representation of $\mathrm{SL}_2(\mathbf{C})$ has $\mathsf{C}(V) = 1$. The logarithm can be genuinely present along families of symmetric and exterior powers of $\mathrm{SL}_n(\mathbf{C})$ with $n$ varying, and it is present for finite Heisenberg groups and $2$-transitive groups of small order such as $\mathrm{PSL}_2(\mathbf{F}_p)$. The complexity is bounded above by a constant on the conjugation representations of $\mathrm{SL}_n(\mathbf{C})$ and on the representations $\operatorname{Sym}^k \mathbf{C}^3$ of $\mathrm{SL}_3(\mathbf{C})$. On the other hand, every fixed connected reductive group has a family of irreducible representations whose complexities tend to at least the dimension of the flag variety. Finally, for the Lie algebra $\mathfrak{sl}_3(\mathbf{C})$ acting on $\operatorname{Sym}^k \mathbf{C}^3$, the monomial diameter with respect to $\operatorname{Sym}^k X$ for a plane $X \leq \mathbf{C}^3$ is optimal, while the corresponding $\mathrm{SL}_3(\mathbf{C})$ diameter is not.

math.RT↗
arXiv2026

Perverse schobers and GKZ systems

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.

math.AG↗
arXiv2026

Comparing the Kirwan and noncommutative resolutions of quotient 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.

math.AG↗
arXiv2026

Catalan numbers, parking functions, permutahedra and noncommutative Hilbert schemes

We find an explicit $S_n$-equivariant bijection between the integral points in a certain zonotope in $\mathbb{R}^n$, 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.

math.CO↗
arXiv2026

On the GKZ discriminant locus

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.

math.AG↗
arXiv2025

Comparison of Cohomological and K-theoretical Hall algebra

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.

math.KT↗
arXiv2024

HMS symmetries of toric boundary divisors

Let $X$ be a projective crepant resolution of a Gorenstein affine toric variety and let $((\mathbb{C}^*)^k,f)$ 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 $(\mathbb{C}^*)^k$. Let $\mathcal{K}_A$ be the so-called "stringy Kähler moduli space" of $X$. We show that $π_1(\mathcal{K}_A)$ acts on the wrapped Fukaya category of $D$. Using results by Gammage - Shende and Zhou, this result implies that $π_1(\mathcal{K}_A)$ acts on $D^b(\operatorname{coh}(\partial X))$ where $\partial X$ is the toric boundary divisor of $X$. We show that the induced action of $π_1(\mathcal{K}_A)$ on $K_0(\operatorname{coh}(\partial X))$ may be extended in a natural way to an action on $K_0(X)$ which corresponds to a GKZ system.

math.AG↗
arXiv2022

On cohomological and K-theoretical Hall algebras of symmetric quivers

We give a brief review of the cohomological Hall algebra CoHA $\mathcal{H}$ and the K-theoretical Hall algebra KHA $\mathcal{R}$ associated to quivers. In the case of symmetric quivers, we show that there exists a homomorphism of algebras (obtained from a Chern character map) $\mathcal{R}\to \hat{\mathcal{H}}^σ$ where $\hat{\mathcal{H}}^σ$ is a Zhang twist of the completion of $\mathcal{H}$. Moreover, we establish the equivalence of categories of ``locally finite'' graded modules $\hat{\mathcal{H}}^σ-{\rm Mod}_{lf}\simeq \mathcal{R}_{\mathbb Q}-{\rm Mod}_{lf}$. Examples of locally finite $\hat{\mathcal{H}}^σ$-, resp. $\mathcal{R}_{\mathbb Q}$- modules appear naturally as the cohomology, resp. K-theory, of framed moduli spaces of quivers.

math.RT↗
arXiv2022

HMS symmetries and hypergeometric systems

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.

math.AG↗
arXiv2020

Associating Geometry to the Lie Superalgebra $\mathfrak{sl}(1|1)$ and to the Color Lie Algebra $\mathfrak{sl}^c_2(\Bbbk)$

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 $\mathfrak{sl}_2(\mathbb{C})$, there is a correspondence between Verma modules and certain line modules that associates a pair $(\mathfrak{h},\,ϕ)$, where $\mathfrak{h}$ is a two-dimensional Lie subalgebra of $\mathfrak{sl}_2(\mathbb{C})$ and $ϕ\in \mathfrak{h}^*$ satisfies $ϕ([\mathfrak{h}, \, \mathfrak{h}]) = 0$, to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra $\mathfrak{sl}(1|1)$ and for a color Lie algebra associated to the Lie algebra $\mathfrak{sl}_2$.

math.RT↗
arXiv2019

A class of perverse schobers in Geometric Invariant Theory

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 established by them. The triangulated categories appearing in our perverse schober are subcategories of the derived category of the quotient stack X/G.

math.AG↗
arXiv2019

Non-commutative crepant resolutions for some toric singularities I

We give a criterion for the existence of non-commutative crepant resolutions (NCCR's) for certain toric singularities. In particular we recover Broomhead's result that a 3-dimensional toric Gorenstein singularity has a NCCR. Our result also yields the existence of a NCCR for a 4-dimensional toric Gorenstein singularity which is known to have no toric NCCR.

math.AG↗
arXiv2019

The Frobenius morphism in invariant theory II

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 $R^{p^r}$.

math.AG↗
arXiv2018

Non-commutative crepant resolutions for some toric singularities II

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.

math.AG↗
arXiv2018

Tilting bundles on hypertoric varieties

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 GIT 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.

math.AG↗