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Benjamin Dequêne

Publications and source records attributed to Benjamin Dequêne.

11 recordsLinked to original sources
arXiv2026

Resolving subcategories for gentle algebras I: Monogeneous resolving subcategories for gentle trees

This paper is the first part of a series that intends to study resolving subcategories for gentle algebras over an algebraically closed field $\mathbb{K}$. In a general setting, we improve the precision of Takahashi's algorithm for resolving closure calculations in well-behaved abelian categories. Then, we modify the geometric model of Baur--Coelho-Simões and Opper--Plamondon--Schroll to compute such subcategories for gentle quivers that have a finite global dimension. Finally, we focus on gentle quivers $(Q,R)$ such that $Q$ is a directed tree, and we study monogeneous resolving subcategories, which are the ones generated by a single non-projective indecomposable $\mathbb{K}Q/\langle R \rangle$-module. Moreover, we prove that these subcategories are the join-irreducible elements of the poset of all resolving subcategories ordered by inclusion.

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arXiv2026

An extended generalization of RSK correspondence via $A$ type quiver representations

Let $λ=(λ_1 \geqslant \ldots \geqslant λ_k > 0)$. For any $c$ Coxeter element of $\mathfrak{S}_{λ_1+k-1}$, we construct a bijection from fillings of $λ$ to reverse plane partitions. We recover two previous generalizations of the Robinson--Schensted--Knuth correspondence for particular choices of Coxeter element depending on $λ$: one based on the work of, among others, Burge, Hillman, Grassl, Knuth, and uniformly presented by Gansner; the other developed by Garver, Partrias, and Thomas, and independently by Dauvergne, called Scrambled RSK. Our results in this paper develop the combinatorial consequence of our previous work of type $A_{λ_1+k}$ quivers.

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arXiv2026

Exact structures and maximal canonically Jordan recoverable subcategories for modules over type $A$ algebras

On one hand, exact structures were introduced by D. Quillen in the '70s. They can be defined as collections of short exact sequences in a fixed abelian category satisfying additional properties. On the other hand, in a recent work, A. Garver, R. Patrias, and H. Thomas introduced Jordan recoverability. Given a bounded quiver $(Q,R)$, a full additive subcategory of $\operatorname{rep}(Q,R)$ is said to be Jordan recoverable if any $X \in \mathscr{C}$ can be recovered, up to isomorphism, from the Jordan form of its generic nilpotent endomorphisms. Such a subcategory $\mathscr{C}$ is said to be canonically Jordan recoverable if, moreover, there exists a precise algebraic procedure that allows one to get back $X \in \mathscr{C}$ from that same Jordan form data. We introduce a new family of operators, called Gen-Sub operators $\operatorname{GS}_\mathcal{E}$, parametrized by the exact structures $\mathcal{E}$ of abelian categories. After showing some properties of those operators in hereditary abelian categories, by focusing on the setting of modules over path algebras of type $A$ quivers endowed with the diamond exact structure $\mathcal{E}_\diamond$, we establish that the maximal canonically Jordan recoverable subcategories are precisely of the form $\operatorname{GS}_{\mathcal{E}_\diamond}(T)$ for some tilting objects $T$.

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arXiv2026

Resolving subcategories for gentle algebras III: Tilting modules for gentle tree algebras

This paper is the third part of a series that intends to study the resolving subcategories for gentle algebras over an algebraically closed field $\mathbb{K}$. As in the previous two papers, we continue to focus on gentle trees $(Q,R)$. Via a modified surface model for gentle algebras with finite global dimension, we developed combinatorial, poset, and quiver representation techniques that allow one to calculate all the resolving subcategories of $\mathbb{K}Q/\langle R \rangle$-mod. Furthermore, they enable one to calculate the resolving subcategory generated by any collection of $\mathbb{K}Q/\langle R \rangle$-modules. In this paper, based on those techniques, we give a combinatorial realization of the Auslander--Reiten one-to-one correspondence between resolving subcategories and tilting modules in $\mathbb{K}Q/\langle R \rangle$-mod.

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arXiv2026

Canonically Jordan recoverable categories for modules over the path algebra of $A_n$ type quivers

Let $Q$ be a quiver of $A_n$ type and $\mathbb{K}$ be an algebraically closed field. A nilpotent endomorphism of a quiver representation induces a linear transformation of the vector space at each vertex. Generically among all nilpotent endomorphisms of a fixed representation $X$, there exists a well-defined Jordan form of each of these linear transformations $\operatorname{GenJF}(X)$, called the generic Jordan form data of $X$. A subcategory of $\operatorname{rep}(Q)$ is Jordan recoverable if we can recover $X$ up to isomorphism from its generic Jordan form data. There is a procedure which allows one to invert the map from representations to generic Jordan form data. The subcategories for which this procedure works are called canonically Jordan recoverable. We focus on the subcategories of $\operatorname{rep}(Q)$ that are canonically Jordan recoverable, and we give a combinatorial characterization of them.

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arXiv2026

Flows on graphs with cycles, locally gentle algebras, and the Mutoperhedron

Flow cones of a directed acyclic graph admit a family of unimodular triangulations given by Danilov, Karzanov, and Koshevoy (DKK) whose normal fans are related to (generalizations) of the associahedron and permutahedron. A correspondence between these triangulations for certain graphs and maximal cones of a $g$-vector fan of a gentle quiver associated to the graph was discovered by von Bell, Braun, Bruegge, Hanely, Peterson, Serhiyenko, and Yip in 2022. This correspondence has been fruitful in uncovering lattice structures in the triangulations. We start by showing that this correspondence is actually a linear isomorphism. We then consider flow cones of certain graphs with cycles. For this case, we give a DKK-like triangulation of the cone, and extend the correspondence to the finite $g$-vector fan of a corresponding locally gentle quiver. In addition, we extend to cyclic graphs a mysterious result of Postnikov--Stanley and Baldoni--Vergne, giving the volume of flow polytopes of acyclic graphs as the number of certain integer flows on the same graph. We illustrate our results with a two-parameter family of cyclic graphs that includes a cycle graph and nested 2-cycles as special cases. We show that the fans of its DKK-like triangulations are respectively isomorphic to the normal fan of the cyclohedron and of a new polytope with the same $f$-vector but different combinatorial type than the permutohedron, which we call the mutoperhedron.

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arXiv2025

Resolving subcategories for gentle algebras II: Resolving subcategories for gentle trees

This paper is the second part of a series that intends to study the resolving subcategories for gentle algebras over an algebraically closed field $\mathbb{K}$. As in the first part, we continue to focus on gentle quivers $(Q,R)$, where $Q$ is a directed tree, known as gentle trees. In our previous work, via a modified surface model for gentle algebras with finite global dimension, we studied the join-irreducible elements of the lattice of resolving subcategories of $\mathbb{K}Q/\langle R \rangle - \text{mod}$, which happen to be those generated by a non-projective indecomposable object. In this paper, we notice that this lattice is not semidistributive in general and, accordingly, introduce a so-called upper join-decomposition, replacing the canonical one. Together with the techniques we develop in our geometric model, it allows us to describe the resolving subcategories of any gentle tree. These same techniques let us explicitly construct the resolving subcategory generated by any collection of indecomposable $\mathbb{K}Q/\langle R\rangle$-modules.

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arXiv2024

An extended generalization of RSK via the combinatorics of type $A$ quiver representations

The classical Robinson--Schensted--Knuth correspondence is a bijection from nonnegative integer matrices to pairs of semi-standard Young tableaux. Based on the work of, among others, Burge, Hillman, Grassl, Knuth and Gansner, it is known that a version of this correspondence gives, for any nonzero integer partition $λ$, a bijection from arbitrary fillings of $λ$ to reverse plane partitions of shape $λ$, via Greene--Kleitman invariants. By bringing out the combinatorial aspects of our recent results on quiver representations, we construct a family of bijections from fillings of $λ$ to reverse plane partitions of shape $λ$ parametrized by a choice of Coxeter element in a suitable symmetric group. We recover the above version of the Robinson--Schensted--Knuth correspondence for a particular choice of Coxeter element depending on $λ$.

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arXiv2024

Jordan recoverability of some subcategories of modules over gentle algebras

Gentle algebras form a class of finite-dimensional algebras introduced by I. Assem and A. Skowroński in the 1980s. Modules over such an algebra can be described by string and band combinatorics in the associated gentle quiver from the work of M.C.R. Butler and C.M. Ringel. Any module can be naturally associated to a quiver representation. A nilpotent endomorphism of a quiver representation induces linear transformations over vector spaces at each vertex. Generically among all nilpotent endomorphisms, a well-defined Jordan form exists for these representations. We focus on subcategories additively generated by all the indecomposable representations of a gentle quiver, including a fixed vertex in their support. We show a characterization of the vertices such that the objects of this subcategory are determined up to isomorphism by their generic Jordan form.

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arXiv2022

Charmed roots and the Kroweras complement

Although both noncrossing partitions and nonnesting partitions are uniformly enumerated for Weyl groups, the exact relationship between these two sets of combinatorial objects remains frustratingly mysterious. In this paper, we give a precise combinatorial answer in the case of the symmetric group: for any standard Coxeter element, we construct an equivariant bijection between noncrossing partitions under the Kreweras complement and nonnesting partitions under a Coxeter-theoretically natural cyclic action we call the Kroweras complement. Our equivariant bijection is the unique bijection that is both equivariant and support-preserving, and is built using local rules depending on a new definition of charmed roots. Charmed roots are determined by the choice of Coxeter element -- in the special case of the linear Coxeter element $(1, 2, \dots, n)$, we recover one of the standard bijections between noncrossing and nonnesting partitions.

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