Calcul des invariants de groupes de permutations par transformée de Fourier

Abstract : This thesis concerns algorithmic approaches to three challenging problems in computational algebraic combinatorics.The firsts parts propose a Gröbner basis free approach for calculating the secondary invariants of a finite permutation group, proceeding by using evaluation at appropriately chosen points. This approach allows for exploiting the symmetries to confine the calculations into a smaller quotient space, which gives a tighter control on the algorithmic complexity, especially for large groups. The theoretical study is illustrated by extensive benchmarks using a fine implementation of algorithms. An important prerequisite is the generation of integer vectors modulo the action of a permutation group, whose algorithmic constitute a preliminary part of the thesis.The fourth part of this thesis is determining for a certain interesting quotient of an affine Hecke algebra exactly which root-of-unity specialization of its parameter lead to non-generic behavior.Finally, the last part presents a conjecture on the structure of certain q-deformed diagonal harmonics in many sets of variables for the infinite family of complex reflection groups.All chapters proceed widely by computer exploration, and most of established algorithms constitute contributions of the software Sage.
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Nicolas Borie. Calcul des invariants de groupes de permutations par transformée de Fourier. Mathématiques générales [math.GM]. Université Paris Sud - Paris XI, 2011. Français. ⟨NNT : 2011PA112294⟩. ⟨tel-00656789⟩



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