arXiv subjects

Benjamin Dupont

Publications and source records attributed to Benjamin Dupont.

7 recordsLinked to original sources

arXiv2025

Pairs of tree dessins, their Shabat polynomials, and monodromy groups

Coverings of the Riemann sphere by itself, ramified over two points, are given by so-called Shabat polynomials. The correspondence between Grothendieck's dessins d'enfants and Belyi maps then implies a bijection between Shabat polynomials and tree dessins (bicolored plane trees). Dessins can be assigned a combinatorial invariant known as their passport, which records the degrees of their vertices. We consider all possible passports determining a pair of tree dessins, determining the associated Shabat polynomials and monodromy groups.

math.AG↗

arXiv2021

Categorification of infinite-dimensional $\mathfrak{sl}_2$-modules and braid group 2-actions I: tensor products

This is the first part of a series of two papers aiming to construct a categorification of the braiding on tensor products of Verma modules, and in particular of the Lawrence--Krammer--Bigelow representations. \\ In this part, we categorify all tensor products of Verma modules and integrable modules for quantum $\mathfrak{sl_2}$. The categorification is given by derived categories of dg versions of KLRW algebras which generalize both the tensor product algebras of Webster, and the dg-algebras used by Lacabanne, the second author and Vaz. We compute a basis for these dgKLRW algebras by using rewriting methods modulo braid-like isotopy, which we develop in an Appendix.

math.QA↗

arXiv2021

Super rewriting theory and nondegeneracy of odd categorified sl(2)

We develop the rewriting theory for monoidal supercategories and 2-supercategories. This extends the theory of higher-dimensional rewriting established for (linear) 2-categories to the super setting, providing a suite of tools for constructing bases and normal forms for 2-supercategories given by generators and relations. We then employ this newly developed theory to prove the non-degeneracy conjecture for the odd categorification of quantum sl(2) from arXiv:1307.7816 and arXiv:1701.04133. As a corollary, this gives a classification of dg-structures on the odd 2-category conjectured in arXiv:1808.04924.

math.QA↗

arXiv2020

Confluence of algebraic rewriting systems

Convergent rewriting systems on algebraic structures give methods to solve decision problems, to prove coherence results, and to compute homological invariants. These methods are based on higher-dimensional extensions of the critical branching lemma that proves local confluence from confluence of the critical branchings. The analysis of local confluence of rewriting systems on algebraic structures, such as groups or linear algebras, is complicated because of the underlying algebraic axioms. This article introduces the structure of algebraic polygraph modulo that formalizes the interaction between the rules of an algebraic rewriting system and the inherent algebraic axioms, and we show a critical branching lemma for algebraic polygraphs. We deduce a critical branching lemma for rewriting systems on algebraic models whose axioms are specified by convergent modulo rewriting systems. We illustrate our constructions for string, linear, and group rewriting systems.

math.CT↗

arXiv2019

Rewriting modulo isotopies in Khovanov-Lauda-Rouquier's categorification of quantum groups

We study a presentation of Khovanov - Lauda - Rouquier's candidate $2$-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to compute a family of linear bases for all the vector spaces of $2$-cells in this $2$-category. We show that these bases correspond to Khovanov and Lauda's conjectured generating sets, proving the non-degeneracy of their diagrammatic calculus. This implies that this $2$-category is a categorification of Lusztig's idempotent and integral quantum group $\bf{U}_{q}(\mathfrak{g})$ associated to a symmetrizable simply-laced Kac-Moody algebra $\mathfrak{g}$.

math.RT↗

arXiv2019

Rewriting modulo isotopies in pivotal linear $(2,2)$-categories

In this paper, we study rewriting modulo a set of algebraic axioms in categories enriched in linear categories, called linear~$(2,2)$-categories. We introduce the structure of linear~$(3,2)$-polygraph modulo as a presentation of a linear~$(2,2)$-category by a rewriting system modulo algebraic axioms. We introduce a symbolic computation method in order to compute linear bases for the vector spaces of $2$-cells of these categories. In particular, we study the case of pivotal $2$-categories using the isotopy relations given by biadjunctions on $1$-cells and cyclicity conditions on $2$-cells as axioms for which we rewrite modulo. By this constructive method, we recover the bases of normally ordered dotted oriented Brauer diagrams in te affine oriented Brauer linear~$(2,2)$-category.

math.CT↗

arXiv2018

Coherent confluence modulo relations and double groupoids

A coherent presentation of an n-category is a presentation by generators, relations and relations among relations. Confluent and terminating rewriting systems generate coherent presentations, whose relations among relations are defined by confluence diagrams of critical branchings. This article introduces a procedure to compute coherent presentations when the rewrite relations are defined modulo a set of axioms. Our coherence results are formulated using the structure of n-categories enriched in double groupoids, whose horizontal cells represent rewriting paths, vertical cells represent the congruence generated by the axioms and square cells represent coherence cells induced by diagrams of confluence modulo. We illustrate our constructions on rewriting systems modulo commutation relations in commutative monoids, isotopy relations in pivotal monoidal categories, and inverse relations in groups.

math.CT↗