arXiv subjects

Léo Schelstraete

Publications and source records attributed to Léo Schelstraete.

6 recordsLinked to original sources

arXiv2026

Monoidal Gröbner systems and categories of affine Brauer type

We introduce an analogue of Gröbner bases, or equivalently, Bergman's diamond lemma, for linear (super)monoidal categories. It gives a systematic way to study monoidal ideals and to prove basis theorems. The central notion is that of a monoidal Gröbner system and is based on higher linear rewriting theory. We then apply the theory to categories of affine Brauer type: linear (super)monoidal categories that have the same hom-basis as the affine Brauer category, but possibly distinct composition and tensor product. Our main result is a criterion for a category to be of affine Brauer type, reducing the question to an explicit list of local computations, which we partially implement in the computer algebra system FORM. This gives new combinatorial proofs of the basis theorems for the affine Brauer category, the nil-Brauer category and the affine VW supercategory, and yields new examples. In particular, we construct the odd nil-Brauer category, a conjectural supercategorification of the split $\imath$quantum group of rank one, and the quantized affine VW supercategory, whose non-affine part is the quantized periplectic Brauer category and which we expect to satisfy a higher quantum Schur--Weyl duality. Finally, we classify the categories of affine Brauer type admitting a sufficiently simple presentation.

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arXiv2026

Odd Khovanov homology and higher representation theory

We introduce a super analogue of $\mathfrak{gl}_2$-foams, and use it to define an invariant of oriented tangles, shown to coincide with odd Khovanov homology when restricted to links. We then define a supercategorification of the $q$-Schur algebra of level 2 and realize our construction as a certain super-2-representation. This gives a representation-theoretic construction of odd Khovanov homology, where signs naturally arise as a byproduct of the super-2-categorical structure. In the process, we define a tensor product for chain complexes in super-2-categories, suitably compatible with homotopies. This could be of independent interest.

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arXiv2026

Rewriting modulo in diagrammatic algebras and application to categorification

We develop a rewriting theory modulo suitable for higher linear structures. In particular, the theory is suited for diagrammatic algebras as they appear in categorification, representation theory and quantum topology. As an application, we use the theory to prove the basis conjecture of a certain super-2-category related to odd Khovanov homology. Our approach combines linear rewriting, higher rewriting and rewriting modulo. For diagrammatic algebras, the modulo rules typically capture a categorical property, such as pivotality. In the process, we revisit the foundations of these theories, including the notion of confluence. Other important tools include termination rules that depend on contexts, rewriting modulo invertible scalars, and a method to classify branchings modulo. This article includes an introduction to rewriting theory for non-experts.

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arXiv2025

A local $\mathfrak{gl}_{1|1}$-action on odd Khovanov homology

We show that odd Khovanov homology carries an action of the super Lie algebra $\mathfrak{gl}_{1|1}$, given extra choice of markings on the link. Moreover, we show that this action arises from an action on super $\mathfrak{gl}_{2}$-foams, in the extended-TQFT framework developed by the second author and Vaz; in particular, it extends to tangles. Finally, we relate the action to torsion $\mathbb{Z}/n\mathbb{Z}$ in pretzel links $P(n,n,-n)$. In particular, this shows that all torsion can appear in odd Khovanov homology.

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arXiv2025

A basis and Schur-Weyl duality for the loop Hecke algebra

The loop Hecke algebra is a generalization of the Hecke algebra to the loop braid group, introduced by Damiani, Martin and Rowell. We give a new presentation of the loop Hecke algebra provided a mild condition on the parameter and give a basis. We use higher linear rewriting theory to show linear independence and the combinatorics of Dyck paths to compute the cardinality of the basis. This yields a conjecture of Damiani-Martin-Rowel. We also give a representation theoretic interpretation of the loop Hecke algebra in terms of (non-semisimple) Schur-Weyl duality involving the negative half of quantum $\mathfrak{gl}_{1|1}$.

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arXiv2024

Odd Khovanov homology, higher representation theory and higher rewriting theory

This thesis is devoted to the fields of quantum topology and rewriting theory, and their surprising interconnections. In the first part of the thesis, we develop a higher representation theoretic approach to odd Khovanov homology; this is the content of arXiv:2311.14394. One of the essential ingredients is a certain graded-2-category of graded $\mathfrak{gl}_2$-foams. In the second part of the thesis, we develop a rewriting theory suitable for higher algebras and their super or graded analogues, and use it to show a basis theorem for graded $\mathfrak{gl}_2$-foams. These techniques have the potential to be applied to a wide variety of contexts. Both parts of the thesis can be read independently. Each has its own comprehensive introduction, allowing experts from one field to get acquainted with the other field.

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