Alg\`ebre combinatoire et effective: des graphes aux alg\`ebres de Kac, via l'exploration informatique

TL;DR

This paper reviews fifteen years of research in algebraic combinatorics, emphasizing computational exploration, invariant theory, and algebraic structures like Kac algebras, with a focus on algorithmic tools and open-source software.

Contribution

It synthesizes research on algebraic combinatorics, introduces computational tools and models, and discusses the development of the open-source *-Combinat project for algebraic exploration.

Findings

Invariant theory applications to graph isomorphism problems

Development of combinatorial models for algebraic structures

Implementation of the *-Combinat software for algebraic computation

Abstract

This manuscript synthesizes almost fifteen years of research in algebraic combinatorics, in order to highlight, theme by theme, its perspectives. In part one, building on my thesis work, I use tools from commutative algebra, and in particular from invariant theory, to study isomorphism problems in combinatorics. I first consider algebras of graph invariants in relation with Ulam's reconstruction conjecture, and then, more generally, the age algebras of relational structures. This raises in return structural and algorithmic problems in the invariant theory of permutation groups. In part two, the leitmotiv is the quest for simple yet rich combinatorial models to describe algebraic structures and their representations. This includes the Hecke group algebras of Coxeter groups which I introduced and which relate to the affine Hecke algebras, but also some finite dimensional Kac algebras in relation with inclusions of factors, and the rational Steenrod algebras. Beside being concrete and constructive, such combinatorial models shed light on certain algebraic phenomena and can lead to elegant and elementary proofs. My favorite tool is computer exploration, and the algorithmic and effective aspects play a major role in this manuscript. In particular, I describe the international open source project *-Combinat which I founded back in 2000, and whose mission is to provide an extensible toolbox for computer exploration in algebraic combinatorics and to foster code sharing among researchers in this area. I present specific challenges that the development of this project raised, and the original algorithmic, design, and development model solutions I was led to develop.