Ph.D
Group : Graphs, ALgorithms and Combinatorics
Representation of monoids and lattice structures in the combinatorics of Weyl groups
Starts on 01/10/2015
Advisor : HIVERT, Florent
Funding : Bourse association
Affiliation : Université Paris-Saclay
Laboratory : LRI - GALaC
Defended on 25/06/2018, committee :
Directeurs de thèse :
Florent Hivert (Université Paris Sud)
Vincent Pilaud (École polytechnique)
Rapporteurs :
Nantel Bergeron (York University)
Riccardo Biagioli (Université Claude Bernard)
Jury :
Nantel Bergeron (York University)
Riccardo Biagioli (Université Claude Bernard)
Marc Baboulin (Université Paris Sud)
Viviane Pons (Université Paris Sud)
Jean-Christophe Novelli (Université Paris Est)
Patrick Dehornoy (Université de Caen)
Florent Hivert (Université Paris Sud)
Vincent Pilaud (École polytechnique)
Research activities :
Abstract :
Algebraic combinatorics is the research field that uses combinatorial methods and algorithms to study algebraic computation, and applies algebraic tools to combinatorial problems.
One of the central topics of algebraic combinatorics is the study of permutations, interpreted in many different ways (as bijections, permutation matrices, words over integers, total orders on integers, vertices of the permutahedron...). This rich diversity of perspectives leads to the following generalizations of the symmetric group. On the geometric side, the symmetric group generated by simple transpositions is the canonical example of finite reflection groups, also called Coxeter groups. On the monoidal side, the simple transpositions become bubble sort operators that generate the $0$-Hecke monoid, whose algebra is the specialization at $q=0$ of Iwahori’s $q$-deformation of the symmetric group. This thesis deals with two further generalizations of permutations.The first one is to introduce a degeneracy at q=0 of Solomon's algebra of the q deformed rook monoid (also called monoid of partial permutations) and to study its representation theory and its equivalent in other Weyl types. The second is to generalize the notion of weak order from permutations to root systems following the recent works of G. Chatel, V. Pilaud and V. Pons.