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Christof Geiß

Publications and source records attributed to Christof Geiß.

At least 19 recordsLinked to original sources
arXiv2025

On homomorphisms and generically $τ$-regular components for skewed-gentle algebras

Let $K$ be an algebraically closed field with $\operatorname{char}(K)\neq 2$, and $A$ a skewed-gentle $K$-algebra. In this case, Crawley-Boevey's description of the indecomposable $A$-modules becomes particularly easy. This allows us to provide an explicit basis for the homomorphisms between any two indecomposable representations in terms of the corresponding admissible words in the sense of Qiu and Zhou. Previously (Geiss, 1999), such a basis was only available when no asymmetric band modules were involved. We also extend a relaxed version of fringing and kisses from Brüstle et al. (2020) to the setting of skewed-gentle algebras. With this at hand, we obtain convenient formulae for the E-invariant and g-vector for indecomposable $A$-modules, similar to the known expressions for gentle algebras. Note however, that we allow in our context also band-modules. As an application, we describe the indecomposable, generically $τ$-regular irreducible components of the representation varieties of $A$ as well as the generic values of the E-invariant between them in terms of tagged admissible words.

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arXiv2025

Bangle functions are the generic basis for cluster algebras from punctured surfaces with boundary

We prove that for any possibly-punctured surface with non-empty boundary $\mathbfΣ=(Σ, \mathbb{M}, \mathbb{P})$, and any tagged triangulation $T$ of $\mathbfΣ$ in the sense of Fomin--Shapiro--Thurston, the coefficient-free bangle functions of Musiker--Schiffler--Williams coincide with the coefficient-free generic Caldero--Chapoton functions arising from the Jacobian algebra of the quiver with potential $(Q(T), W(T))$ associated to $T$ by Cerulli Irelli and the second author. When the set of boundary marked points $\mathbb{M}$ has at least two elements, Schröer and the first two authors have shown, relying heavily on results of Mills, Muller and Qin, that the generic coefficient-free Caldero-Chapoton functions form a basis of the coefficient-free (upper) cluster algebra $\mathcal{A}(\mathbfΣ)=\mathcal{U}(\mathbfΣ)$. So, the set of bangle functions proposed by Musiker--Schiffler--Williams over ten years ago is indeed a basis.

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arXiv2025

Laminations of punctured surfaces as $τ$-regular irreducible components

Let $\boldsymbolΣ:=(Σ,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subset\partialΣ\neq\varnothing$ on the boundary, and punctures $\mathbb{P}\subsetΣ\setminus\partialΣ$, and $T$ an arbitrary tagged triangulation of $\boldsymbolΣ$ in the sense of Fomin-Shapiro-Thurston. The Jacobian algebra $A(T):=\mathcal{P}(Q(T), W(T))$ corresponding to the non-degenerate potential $W(T)$ defined by Cerulli Irelli and the second author is tame, as shown by Schröer and the first two authors. In this paper, we show that there is a natural isomorphism $π_T:\operatorname{Lam}(\boldsymbolΣ)\rightarrow\operatorname{DecIrr}^τ(A(T))$ of tame partial KRS-monoids that intertwines dual shear coordinates with respect to $T$, and generic $g$-vectors of irreducible components. Here, $\operatorname{Lam}(\boldsymbolΣ)$ is the set of laminations of $\boldsymbolΣ$ considered by Musiker-Schiffler-Williams, with the disjoint union of non-intersecting laminations as partial monoid operation. On the other hand, $\operatorname{DecIrr}^τ(A(T))$ denotes the set of generically $τ$-regular irreducible components of the decorated representation varieties of $A(T)$, with the direct sum of generically $E$-orthogonal irreducible components as partial monoid operation, where $E$ is the symmetrized $E$-invariant of Derksen-Weyman-Zelevinsky, $E(-,\bullet)=\dim\operatorname{Hom}_{A(T)}(-,τ(\bullet))+\dim\operatorname{Hom}_{A(T)}(\bullet,τ(-))$.

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arXiv2024

A model for the canonical algebras of bimodules type (1, 4) over truncated polynomial rings

Let $k=\mathbb{C}(\!(ε)\!)$ be the field of complex Laurent series. We use Galois descent techniques to show that the simple regular representations of the species of type $(1,\, 4)$ over $k$ are naturally parametrized by the closed points of $\mathrm{Spec}(k[x])\dot{\cup}\{1,\,2\}$. Moreover we provide weak normal forms for those representations. We use our representatives of the simple regular representations to describe the canonical algebras associated to the species of type (1, 4) over k. This suggest a model of those algebras in the sense of the work of Geiss, Leclerc and Schröer [GLS17] and [GLS20].

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arXiv2024

Semicontinuous maps on module varieties

We study semicontinuous maps on varieties of modules over finite-dimensional algebras. We prove that truncated Euler maps are upper or lower semicontinuous. This implies that $g$-vectors and $E$-invariants of modules are upper semicontinuous. We also discuss inequalities of generic values of some upper semicontinuous maps.

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arXiv2024

On the resolution of kinks of curves on punctured surfaces

Let $(Σ,\mathbb{M},\mathbb{P})$ be a surface with marked points $\mathbb{M}\subseteq \partialΣ\neq\varnothing$ and punctures $\mathbb{P}\subseteqΣ\setminus\partialΣ$. In this paper we show that for every curve $γ$ on $Σ\setminus\mathbb{P}$, the curve obtained by resolving the kinks of $γ$ in any order is uniquely determined, up to homotopy in $Σ\setminus\mathbb{P}$, by the $2$-orbifold homotopy class of $γ$, in which the punctures are interpreted to be orbifold points of order $2$. Our proof resorts to an application of the Diamond Lemma.

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arXiv2021

Schemes of modules over gentle algebras and laminations of surfaces

We study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by dropping acyclicity, and by incorporating band modules. A special class of gentle algebras are Jacobian algebras arising from triangulations of unpunctured marked surfaces. For these we obtain a bijection between the set of generically tau-reduced decorated irreducible components and the set of laminations of the surface. As an application, we get that the set of bangle functions (defined by Musiker-Schiffler-Williams) in the upper cluster algebra associated with the surface coincides with the set of generic Caldero-Chapoton functions.

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arXiv2020

Generic Caldero-Chapoton functions with coefficients and applications to surface cluster algebras

We realize Derksen-Weyman-Zelevinsky's mutations of representations as densely-defined regular maps on representation spaces, and study the generic values of Caldero-Chapoton functions with coefficients, giving, for instance, a sufficient combinatorial condition for their linear independence. For a quiver with potential $(Q,S)$, we show that if $k$ is a vertex not incident to any oriented 2-cycle, then every generically $τ$-reduced irreducible component $Z$ of any affine variety of (decorated) representations has a dense open subset $U$ on which Derksen-Weyman-Zelevinsky's mutation of representations $μ_k$ can be defined consistently as a regular map to an affine variety of (decorated) representations of the Jacobian algebra of the mutated QP $μ_k(Q,S)$. Our techniques involve only basic linear algebra and elementary algebraic geometry, and do not require to assume Jacobi-finiteness. Thus, the paper yields a new and more general proof of the mutation invariance of generic Caldero-Chapoton functions, generalizing and providing a new natural geometric perspective on results of Derksen-Weyman-Zelevinsky and Plamondon. (For Jacobi-finite non-degenerate quivers with potential, this invariance was shown by Plamondon using the machinery of Ginzburg dg-algebras and Hom-finite generalized cluster categories.) We apply our results, together with results of Mills, Muller and Qin, to prove that for any choice of geometric coefficient systems, not necessarily of full rank, the cluster algebra associated to a possibly punctured surface with at least two marked points on the boundary has the generic Caldero-Chapoton functions as a basis over the Laurent polynomial ring of coefficients.

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arXiv2019

Rigid modules and Schur roots

Let $C$ be a symmetrizable generalized Cartan matrix with symmetrizer $D$ and orientation $Ω$. In previous work we associated an algebra $H$ to this data, such that the locally free $H$-modules behave in many aspects like representations of a hereditary algebra $\tilde{H}$ of the corresponding type. We define a Noetherian algebra $\hat{H}$ over a power series ring, which provides a direct link between the representation theory of $H$ and of $\tilde{H}$. We define and study a reduction and a localization functor relating the module categories of these three types of algebras. These are used to show that there are natural bijections between the sets of isoclasses of tilting modules over $H$, $\hat{H}$ and $\tilde{H}$. We show that the indecomposable rigid locally free modules over $H$ and $\hat{H}$ are parametrized, via their rank vector, by the real Schur roots associated to $(C,Ω)$. Moreover, the left finite bricks of $H$, in the sense of Asai, are parametrized, via their dimension vector, by the real Schur roots associated to the dual datum $(C^T,Ω)$.

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arXiv2019

Quantum cluster algebras and their specializations

We show that in case a cluster algebra coincides with its upper cluster algebra and the cluster algebra admits a grading with finite dimensional homogeneous components, the corresponding Berenstein-Zelevinsky quantum cluster algebra can be viewed as a flat deformation of the classical cluster algebra.

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arXiv2018

Quivers with relations for symmetrizable Cartan matrices IV: Crystal graphs and semicanonical functions

We generalize Lusztig's nilpotent varieties, and Kashiwara and Saito's geometric construction of crystal graphs from the symmetric to the symmetrizable case. We also construct semicanonical functions in the convolution algebras of generalized preprojective algebras. Conjecturally these functions yield semicanonical bases of the enveloping algebras of the positive part of symmetrizable Kac-Moody algebras.

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arXiv2018

Quivers with relations for symmetrizable Cartan matrices V. Caldero-Chapoton formula

We generalize the Caldero-Chapoton formula for cluster algebras of finite type to the skew-symmetrizable case. This is done by replacing representation categories of Dynkin quivers by categories of locally free modules over certain Iwanaga-Gorenstein algebras introduced in Part I. The proof relies on the realization of the positive part of the enveloping algebra of a simple Lie algebra of the same finite type as a convolution algebra of constructible functions on representation varieties of $H$, given in Part III. Along the way, we obtain a new result on the PBW basis of this convolution algebra.

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arXiv2016

The representation type of Jacobian algebras

We show that the representation type of the Jacobian algebra P(Q,S) associated to a 2-acyclic quiver Q with non-degenerate potential S is invariant under QP-mutations. We prove that, apart from very few exceptions, P(Q,S) is of tame representation type if and only if Q is of finite mutation type. We also show that most quivers Q of finite mutation type admit only one non-degenerate potential up to weak right equivalence. In this case, the isomorphism class of P(Q,S) depends only on Q and not on S.

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arXiv2012

Factorial cluster algebras

We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky's Laurent phenomenon. As an application we give a criterion for a cluster algebra to be a factorial algebra. This can be used to construct cluster algebras, which are isomorphic to polynomial rings. We also study various kinds of upper bounds for cluster algebras, and we prove that factorial cluster algebras coincide with their upper bounds.

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arXiv2011

Generic bases for cluster algebras and the Chamber Ansatz

Let Q be a finite quiver without oriented cycles, and let $Λ$ be the corresponding preprojective algebra. Let g be the Kac-Moody Lie algebra with Cartan datum given by Q, and let W be its Weyl group. With w in W is associated a unipotent cell N^w of the Kac-Moody group with Lie algebra g. In previous work we proved that the coordinate ring \C[N^w] of N^w is a cluster algebra in a natural way. A central role is played by generating functions \vphi_X of Euler characteristics of certain varieties of partial composition series of X, where X runs through all modules in a Frobenius subcategory C_w of the category of nilpotent $Λ$-modules. We show that for every X in C_w, \vphi_X coincides after appropriate changes of variables with the cluster characters of Fu and Keller associated with any cluster-tilting module T of C_w. As an application, we get a new description of a generic basis of the cluster algebra obtained from \C[N^w] via specialization of coefficients to 1. For the special case of coefficient-free acyclic cluster algebras this proves a conjecture by Dupont.

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arXiv2007

Semicanonical bases and preprojective algebras II: A multiplication formula

Let $n$ be a maximal nilpotent subalgebra of a complex symmetric Kac-Moody Lie algebra. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of nilpotent modules over a preprojective algebra of the same type as $n$. We prove a formula for the product of two elements of the dual of this semicanonical basis, and more generally for the product of two evaluation forms associated to arbitrary modules over the preprojective algebra. This formula plays an important role in our work on the relationship between semicanonical bases, representation theory of preprojective algebras, and Fomin and Zelevinsky's theory of cluster algebras. It was inspired by recent results of Caldero and Keller.

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arXiv2006

Auslander algebras and initial seeds for cluster algebras

Let $Q$ be a Dynkin quiver and $Π$ the corresponding set of positive roots. For the preprojective algebra $Λ$ associated to $Q$ we produce a rigid $Λ$-module $I_Q$ with $r=|Π|$ pairwise non-isomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of $kQ$ to $Λ$. If $N$ is a maximal unipotent subgroup of a complex simply connected simple Lie group of type $|Q|$, then the coordinate ring $C[N]$ is an upper cluster algebra. We show that the elements of the dual semicanonical basis which correspond to the indecomposable direct summands of $I_Q$ coincide with certain generalized minors which form an initial cluster for $C[N]$, and that the corresponding exchange matrix of this cluster can be read from the Gabriel quiver of $End_Λ(I_Q)$. Finally, we exploit the fact that the categories of injective modules over $Λ$ and over its covering $\tildeΛ$ are triangulated in order to show several interesting identities in the respective stable module categories.

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