Algèbre combinatoire et effective: des graphes aux algèbres de Kac, via l'exploration informatique - Mathematics > CombinatoricsReport as inadecuate




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Abstract: This manuscript synthesizes almost fifteen years of research in algebraiccombinatorics, in order to highlight, theme by theme, its perspectives.In part one, building on my thesis work, I use tools from commutativealgebra, and in particular from invariant theory, to study isomorphism problemsin combinatorics. I first consider algebras of graph invariants in relationwith Ulam-s reconstruction conjecture, and then, more generally, the agealgebras of relational structures. This raises in return structural andalgorithmic problems in the invariant theory of permutation groups.In part two, the leitmotiv is the quest for simple yet rich combinatorialmodels to describe algebraic structures and their representations. Thisincludes the Hecke group algebras of Coxeter groups which I introduced andwhich relate to the affine Hecke algebras, but also some finite dimensional Kacalgebras in relation with inclusions of factors, and the rational Steenrodalgebras. Beside being concrete and constructive, such combinatorial modelsshed light on certain algebraic phenomena and can lead to elegant andelementary proofs.My favorite tool is computer exploration, and the algorithmic and effectiveaspects play a major role in this manuscript. In particular, I describe theinternational open source project *-Combinat which I founded back in 2000, andwhose mission is to provide an extensible toolbox for computer exploration inalgebraic combinatorics and to foster code sharing among researchers in thisarea. I present specific challenges that the development of this projectraised, and the original algorithmic, design, and development model solutions Iwas led to develop.



Author: Nicolas M. Thiéry

Source: https://arxiv.org/







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