Algèbre combinatoire et effective: des graphes aux algèbres de Kac, via l'exploration informatique

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.

Mémoire de Diplôme d'habilitation à diriger des recherche de l'Université Paris Sud, soutenu le 10 Décembre 2008. 80 pages, 14 figures

References in corpus (9)

• A Galois correspondence for II_1 factors and quantum groupoids
• The peak algebra and the Hecke-Clifford algebras at $q=0$
• Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n
• Minimal generating sets of non-modular invariant rings of finite groups
• Hecke group algebras as quotients of affine Hecke algebras at level 0
• Catalan paths, Quasi-symmetric functions and Super-Harmonic Spaces
• Some relational structures with polynomial growth and their associated algebras
• Representation theories of some towers of algebras related to the symmetric groups and their Hecke algebras
• The Algebra of Graph Invariants - Lower and Upper Bounds for Minimal Generators