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Baptiste Rognerud

Publications and source records attributed to Baptiste Rognerud.

16 recordsLinked to original sources
arXiv2026

The extra slow Tamari lattice

We introduce the extra slow Tamari lattices, a new family of lattices defined on faithfully balanced tableaux. These tableaux arise naturally from the representation theory of type \( A \) quivers, and our construction extends the classical Tamari lattice and the slow Tamari lattice. We explicitly describe meets and joins in the extra slow Tamari lattices, and then prove that they are lattices. We then show that they are semidistributive, trim, polygonal, and congruence uniform. Their join-irreducible elements are described in terms of a three-color analogue of the positive roots of type \( A \), which leads to descriptions of their spines and congruence lattices. We also obtain several enumerative results for the extra slow Tamari lattices and their associated structures. Finally, we derive new structural and enumerative results for the slow Tamari lattices.

math.CO↗
arXiv2024

A remark on s-torsion pairs and on the lattice of Dyck paths

There are three classical lattices on the Catalan numbers: the Tamari lattice, the lattice of noncrossing partitions and the lattice of Dyck paths. The first is known to be isomorphic to the lattice of torsion classes of the path algebra of an equioriented quiver of type $A$ and the second is known to be isomorphic to its lattice of wide subcategories. Inspired by the notion of s-torsion classes of Adachi, Enomoto and Tsukamoto, in this short note we interpret the lattice of Dyck paths as a lattice of subcategories.

math.CO↗
arXiv2024

On the lattice of the weak factorization systems on a finite lattice

We consider the lattice of all the weak factorization systems on a given finite lattice. We prove that it is semidistributive, trim and congruence uniform. We deduce a graph theoretical approach to the problem of enumerating transfer systems. As an application we find a lower bound for the number of transfer systems on a boolean lattice.

math.CO↗
arXiv2023

Odd and even Fibonacci lattices arising from a Garside monoid

We study two families of lattices whose number of elements are given by the numbers in even (respectively odd) positions in the Fibonacci sequence. The even Fibonacci lattice arises as the lattice of simple elements of a Garside monoid partially ordered by left-divisibility, and the odd Fibonacci lattice is an order ideal in the even one. We give a combinatorial proof of the lattice property, relying on a description of words for the Garside element in terms of Schröder trees, and on a recursive description of the even Fibonacci lattice. This yields an explicit formula to calculate meets and joins in the lattice. As a byproduct we also obtain that the number of words for the Garside element is given by a little Schröder number.

math.CO↗
arXiv2021

Combinatorics of quasi-hereditary structures

A quasi-hereditary algebra is an Artin algebra together with a partial order on its set of isomorphism classes of simple modules which satisfies certain conditions. In this article we investigate all the possible choices that yield to quasi-hereditary structures on a given algebra, in particular we introduce and study what we call the poset of quasi-hereditary structures. Our techniques involve certain quiver decompositions and idempotent reductions. For a path algebra of Dynkin type $\mathbb{A}$, we provide a full classification of its quasi-hereditary structures. For types $\mathbb{D}$ and $\mathbb{E}$, we give a counting method for the number of quasi-hereditary structures. In the case of a hereditary incidence algebra, we present a necessary and sufficient condition for its poset of quasi-hereditary structures to be a lattice.

math.RT↗
arXiv2019

Combinatorics of faithfully balanced modules

We study and classify faithfully balanced modules for the algebra of lower triangular $n$ by $n$ matrices. The theory extends known results about tilting modules, which are classified by binary trees, and counted with the Catalan numbers. The number of faithfully balanced modules is a $2$-factorial number. Among them are $n!$ modules with $n$ indecomposable summands, which can be classified by interleaved binary trees or by increasing binary trees.

math.RT↗
arXiv2018

The bounded derived categories of the Tamari lattices are fractionally Calabi-Yau

We prove that the bounded derived category of the incidence algebra of the Tamari lattice is fractionally Calabi-Yau, giving a positive answer to a conjecture of Chapoton. The proof involves a combinatorial description of the Serre functor of this derived category on a sufficiently nice family of indecomposable objects.

math.RT↗
arXiv2018

Around evaluations of biset functors

For a non-vanishing group, we show that the evaluation functor induces an equivalence between the category of modules over the double Burnside algebra and a certain category of biset functors. Using this equivalence, we deduce that over a field of characteristic zero, the double Burnside algebra of a non-vanishing group is a quasi-hereditary algebra. We also show that the double Burnside algebra over a field is self-injective if and only if it is semisimple.

math.RT↗
arXiv2018

On Morita and derived equivalences for cohomological Mackey algebras

By results of the second author, a source algebra equivalence between two $p$-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories of cohomological Mackey functors. The main result of this paper proves a partial converse: an equivalence (resp. Rickard equivalence) between the categories of cohomological Mackey functors of two blocks of finite groups induces a permeable Morita (resp. derived) equivalence between the two block algebras.

math.RT↗
arXiv2018

Exceptional and modern intervals of the Tamari lattice

In this article we use the theory of interval-posets recently introduced by Ch{â}tel and Pons in order to describe some interesting families of intervals in the Tamari lattices. These families are defined as interval-posets avoiding specific configurations. At first, we consider what we call exceptional interval-posets and show that they correspond to the intervals which are obtained as images of noncrossing trees in the Dendriform operad. We also show that the exceptional intervals are exactly the intervals of the Tamari lattice induced by intervals in the poset of noncrossing partitions. In the second part we introduce the notion of modern and infinitely modern interval-posets. We show that the modern intervals are in bijection with the new intervals of the Tamari lattice in the sense of Chapoton. We deduce an intrinsic characterization of the new intervals in the Tamari lattice. Finally, we consider the family of what we call infinitely modern intervals and we we prove that there are as many infinitely modern interval-posets of size n as there are ternary trees with n inner vertices.

math.CO↗
arXiv2017

On the wildness of cambrian lattices

In this note, we investigate the representation type of the cambrian lattices and some other related lattices. The result is expressed as a very simple trichotomy. When the rank of the underlined Coxeter group is at most 2, the lattices are of finite representation type. When the Coxeter group is a reducible group of type A 3 1 , the lattices are of tame representation type. In all the other cases they are of wild representation type.

math.RT↗
arXiv2015

Quasi-hereditary property of double Burnside algebras

In this short note we investigate some consequences of the vanishing of simple biset functors. As corollary, if there is no non-trivial vanishing of simple biset functors (e.g. if the group is commutative), then we show that the double Burnside algebra is a quasi-hereditary algebra in characteristic zero. In general, this not true without the non-vanishing condition, as over a field of characteristic zero, the double Burnside algebra of the alternating group of degree 5 has infinite global dimension.

math.RT↗
arXiv2014

Equivalences between blocks of p-local Mackey algebras

Let $G$ be a finite group and $(K,\mathcal{O},k)$ be a $p$-modular system. Let $R=\mathcal{O}$ or $k$. There is a bijection between the blocks of the group algebra and the blocks of the so-called $p$-local Mackey algebra $μ_{R}^{1}(G)$. Let $b$ be a block of $RG$ with abelian defect group $D$. Let $b'$ be its Brauer correspondant in $N_{G}(D)$. It is conjectured by Broué that the blocks $RGb$ and $RN_{G}(D)b'$ are derived equivalent. Here we look at equivalences between the corresponding blocks of $p$-local Mackey algebras. We prove that an analogue of the Broué's conjecture is true for the $p$-local Mackey algebras in the following cases: for the principal blocks of $p$-nilpotent groups and for blocks with defect $1$. We also point out the probable importance of \emph{splendid} equivalences for the Mackey algebras.

math.RT↗
arXiv2014

Equivalences between blocks of cohomological Mackey algebras

Let $G$ be a finite group and $(K,\mathcal{O},k)$ be a $p$-modular system "large enough". Let $R=\mathcal{O}$ or $k$. There is a bijection between the blocks of the group algebra $RG$ and the central primitive idempotents (the blocks) of the so-called cohomological Mackey algebra $coμ_{R}(G)$. Here, we prove that a so-called permeable derived equivalence between two blocks of group algebras implies the existence of a derived equivalence between the corresponding blocks of cohomological Mackey algebras. In particular, in the context of Broué's abelian defect group conjecture, if two blocks are splendidly derived equivalent, then the corresponding blocks of cohomological Mackey algebras are derived equivalent.

math.RT↗
arXiv2014

Trace maps for Mackey algebras

Let $G$ be a finite group and $R$ be a commutative ring. The Mackey algebra $μ_{R}(G)$ shares a lot of properties with the group algebra $RG$ however, there are some differences. For example, the group algebra is a symmetric algebra and this is not always the case for the Mackey algebra. In this paper we present a systematic approach to the question of the symmetricity of the Mackey algebra, by producing symmetric associative bilinear forms for the Mackey algebra. The category of Mackey functors is a closed symmetric monoidal category, so using the formalism of J.P. May for these categories, S. Bouc has defined the so-called Burnside trace. Using this Burnside trace we produce trace maps for Mackey algebras which generalize the usual trace map the group algebras. These trace maps factorise through Burnside algebras. We prove that the Mackey algebra $μ_{R}(G)$ is a symmetric algebra if and only if the family of Burnside algebras $(RB(H))_{H\leqslant G}$ is a family of symmetric algebras with a compatibility condition. As a corollary, we recover the well known fact that over a field of characteristic zero, the Mackey algebra is always symmetric. Over the ring of integers the Mackey algebra of $G$ is symmetric if and only if the order of $G$ is square free. Finally, over a field of characteristic $p>0$ we show that the Mackey algebra is symmetric if and only if the Sylow $p$-subgroups of $G$ are of order $1$ or $p$.

math.GR↗