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Study of plactic monoids by rewriting methods

Abstract : This thesis focuses on the study of plactic monoids by a new approach using methods issued from rewriting theory. These methods are applied on presentations of plactic monoids given in terms of Young tableaux, Kashiwara’s crystal bases and Littelmann path model. We study the syzygy problem for the Knuth presentation of the plactic monoids. Using the homotopical completion procedure that extends Squier’s and Knuth–Bendix’s completions procedure, we construct coherent presentations of plactic monoids of type A. Such a coherent presentation extends the notion of a presentation of a monoid by a family of generating syzygies, taking into account all the relations among the relations. We make explicit a finite coherent presentation of plactic monoids of type A with the column generators. However, this presentation is not minimal in the sense that many of its generators are superfluous. After applying the homotopical reduction procedure on this presentation, we reduce it to a finite coherent one that extends the Knuth presentation, giving then all the syzygies of the Knuth relations. More generally, we deal with presentations of plactic monoids of any type from the rewriting theory perspective. We construct finite convergent presentations for these monoids in a general way using Littelmann paths. Moreover, we study the latter presentations in terms of Kashiwara’s crystal graphs for type C. By introducing the admissible column generators, we obtain a finite convergent presentation of the plactic monoid of type C with explicit relations. This approach should allow us to study the syzygy problem for the presentations of plactic monoids for any type
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  • HAL Id : tel-02071461, version 1


Nohra Hage. Study of plactic monoids by rewriting methods. Combinatorics [math.CO]. Université de Lyon, 2016. English. ⟨NNT : 2016LYSES065⟩. ⟨tel-02071461⟩



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