arXiv subjects

Alexis Langlois-Rémillard

Publications and source records attributed to Alexis Langlois-Rémillard.

13 recordsLinked to original sources

arXiv2026

Maximum independent queen set on polyominoes is NP-complete

Finding a set of vertices in a graph with no edges between them, INDSET, is a well-known NP-complete problem. The queen graph of a chessboard is constructed by taking vertices as the tiles of the chessboard and drawing edges between two tiles if a queen can move from one to the other. We call INDQUEENS the independent set problem on a queen graph where the chessboard is a polyomino. We prove that INDQUEENS on polyominoes is NP-complete, proving a conjecture of Langlois-R\'emillard--M\"u{\ss}ig--Rold\'an. As our reduction is parsimonious, we can further prove that it is #P-complete. We furthermore prove that INDROOKS on polyominoes is #P-complete, despite being solvable in polynomial time.

cs.CC↗

arXiv2026

Extremal fences with polyforms

We present results around an isoperimetric problem built on polyforms: What is the biggest enclosed area one can build using polyforms in each of the three plane tessellations? We give challenges to the readers and present Shimauchi's proof of the biggest area a fence made of pentominoes can enclose. A translation of the instance using integer linear programming is also given. A companion web app is available to test some of the challenges we propose and for activities, and we included extra pages with a cutout handout of the board and pieces of the puzzles, so that you can print and cut them to read this paper hands on.

math.HO↗

arXiv2026

Diagrammatics for lax and Frobenius monoidal functors and weak morphism classifiers

The theory of 2-monads entails that, for a strict monoidal category C, there is a strict monoidal category L(C) such that strict monoidal functors from L(C) are precisely the lax monoidal functors from C. We give an elementary, diagrammatic, construction of L(C) and of its variants for oplax and Frobenius lax functors. The diagrams used are analogous to the diagrammatics for lax monoidal functors studied by McCurdy.

math.CT↗

arXiv2024

Insights from a workshop on gamification of research in mathematics and computer science

Can outreach inspire and lead to research and vice versa? In this work, we introduce our approach to the gamification of research in mathematics and computer science through three illustrative examples. We discuss our primary motivations and provide insights into what makes our proposed gamification effective for three research topics in discrete and computational geometry and topology: (1) DominatriX, an art gallery problem involving polyominoes with rooks and queens; (2) Cubical Sliding Puzzles, an exploration of the discrete configuration spaces of sliding puzzles on the $d$-cube with topological obstructions; and (3) The Fence Challenge, a participatory isoperimetric problem based on polyforms. Additionally, we report on the collaborative development of the game Le Carr\'e du Diable, inspired by The Fence Challenge and created during the workshop Let's talk about outreach!, held in October 2022 in Les Diablerets, Switzerland. All of our outreach encounters and creations are designed and curated with an inclusive culture and a strong commitment to welcoming the most diverse audience possible.

math.HO↗

arXiv2023

Generalised symmetries and bases for Dunkl monogenics

We introduce a family of commuting generalised symmetries of the Dunkl--Dirac operator inspired by the Maxwell construction in harmonic analysis. We use these generalised symmetries to construct bases of the polynomial null-solutions of the Dunkl--Dirac operator. These polynomial spaces form representation spaces of the Dunkl--Dirac symmetry algebra. For the $\mathbb{Z}_2^d$ case, the results are compared with previous investigations.

math.RT↗

arXiv2023

The double dihedral Dunkl total angular momentum algebra

The Dunkl total angular momentum algebra (TAMA) is realised as the dual partner of the orthosymplectic Lie superalgebra containing the Dunkl deformation of the Dirac operator. In this paper, we consider the case when the reflection group associated with the Dunkl operators is a product of two dihedral groups acting on a four-dimensional Euclidean space. We show that in this case there is a subalgebra of the total angular momentum algebra that admits a triangular decomposition. In analogy to the celebrated theory of semisimple Lie algebras, we use this triangular subalgebra to give precise necessary conditions that a finite-dimensional irreducible representation must obey, in terms of weights. In specific cases, which includes unitary representations, we construct a basis of weight vectors with explicit actions of all TAMA elements. Examples of these modules occur in the kernel of the Dunkl--Dirac operator in the context of deformations of Howe dual pairs.

math.RT↗

arXiv2023

Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors

Affine and periodic Temperley-Lieb algebras are families of diagrammatic algebras that find diverse applications in mathematics and physics. These algebras are infinite dimensional, yet most of their interesting modules are finite. In this paper, we introduce finite quotients for these algebras, which we term uncoiled affine Temperley-Lieb algebras and uncoiled periodic Temperley-Lieb algebras. We study some of their properties, including their defining relations, their description with diagrams, their dimensions, and their relations with affine and skew sandwich cellular algebras. The uncoiled algebras all have finitely many one-dimensional modules. We construct a family of Wenzl-Jones idempotents, each of which projects onto one of these one-dimensional modules. Our construction is explicit and uses the similar projectors for the ordinary Temperley--Lieb algebras, as well as the diagrammatic description of the uncoiled algebras in terms of sandwich diagrams. We also discuss the Markov traces for the uncoiled algebras and their evaluations on the newly defined projectors, and find expressions involving Chebyshev polynomials of the first kind.

math.RT↗

arXiv2022

Complexity of chess domination problems

We study different domination problems of attacking and non-attacking rooks and queens on polyominoes and polycubes of all dimensions. Our main result proves that maximum independent domination is NP-complete for non-attacking queens and for non-attacking rooks on polycubes of dimension three and higher. We also analyze these problems for polyominoes and convex polyominoes, conjecture the complexity classes, and provide a computer tool for investigation. We have also computed new values for classical queen domination problems on chessboards (square polyominoes). For our computations, we have translated the problem into an integer linear programming instance. Finally, using this computational implementation and the game engine Godot, we have developed a video game of minimum domination of queens and rooks on randomly generated polyominoes.

math.CO↗

arXiv2022

The dihedral Dunkl--Dirac symmetry algebra with negative Clifford signature

The Dunkl--Dirac symmetry algebra is an associative subalgebra of the tensor product of a Clifford algebra and the faithful polynomial representation of a rational Cherednik algebra. In previous work, the finite-dimensional representations of the Dunkl--Dirac symmetry algebra in three dimensions linked with a dihedral group were given. We give here the necessary results to proceed to the same construction when the Clifford algebra in the tensor product has negative signature.

math.RT↗

arXiv2021

Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl--Dirac operator

The Dunkl--Dirac operator is a deformation of the Dirac operator by means of Dunkl derivatives. We investigate the symmetry algebra generated by the elements supercommuting with the Dunkl--Dirac operator and its dual symbol. This symmetry algebra is realised inside the tensor product of a Clifford algebra and a rational Cherednik algebra associated with a reflection group or root system. For reducible root systems of rank three, we determine all the irreducible finite-dimensional representations and conditions for unitarity. Polynomial solutions of the Dunkl--Dirac equation are given as a realisation of one family of such irreducible unitary representations.

math.RT↗

arXiv2021

Deforming algebras with anti-involution via twisted associativity

This contribution studies a specific deformation of algebras with anti-involution. Starting with the observation that twisting the multiplication of such an algebra by its anti-involution generates a Hom-associative algebra of type II, it formulates the adequate modules theory over these algebras, and shows that there is a faithful functor from the category of finite-dimensional left modules of algebras with involution to finite-dimensional right modules of Hom-associative algebras of type II. It ends with a discussion on a diagrammatic operadic approach to generalize the study of such deformations.

math.RA↗

arXiv2020

An exceptional symmetry algebra for the 3D Dirac-Dunkl operator

We initiate the study of an algebra of symmetries for the 3D Dirac-Dunkl operator associated with the Weyl group of the exceptional root system $G_2$. For this symmetry algebra, we give both an abstract definition and an explicit realisation. We then construct ladder operators, using an intermediate result we prove for the Dirac-Dunkl symmetry algebra associated with arbitrary finite reflection group acting on a three-dimensional space.

math-ph↗

arXiv2019

The representation theory of seam algebras

The boundary seam algebras $\mathsf{b}_{n,k}(β=q+q^{-1})$ were introduced by Morin-Duchesne, Ridout and Rasmussen to formulate algebraically a large class of boundary conditions for two-dimensional statistical loop models. The representation theory of these algebras $\mathsf{b}_{n,k}(β=q+q^{-1})$ is given: their irreducible, standard (cellular) and principal modules are constructed and their structure explicited in terms of their composition factors and of non-split short exact sequences. The dimensions of the irreducible modules and of the radicals of standard ones are also given. The methods proposed here might be applicable to a large family of algebras, for example to those introduced recently by Flores and Peltola, and Crampé and Poulain d'Andecy.

math-ph↗