• 0-Hecke algebra
• algebraic combinatorics
• coinvariant algebra
• representation theory
• symmetric group

In this thesis we study a deformation of the group algebra of the symmetric group, the 0-Hecke algebra. After recalling some preliminaries, we define this algebra by modifying the Coxeter presentation of the group algebra of the symmetric group. The 0-Hecke algebra is not a semisimple algebra, thus we cannot hope to decompose every module in simple ones. In order to find a substitute to the notion of simple module, we define the composition series and the indecomposable projective modules, and we relate them through the notions of top and projective cover of a module. We introduce the algebra of non-commutative symmetric functions and the algebra of quasisymmetric functions that play the role of the algebra of the symmetric functions. In fact, there exist two isomorphisms of algebras, the Hecke-Frobenius-Schur isomorphisms, that play the same role as the Frobenius-Schur isomorphism, creating a bridge between modules and functions, in order to make possible a dual approach to the study of these objects. Finally, we study an action of the 0-Hecke algebra on the algebra of coinvariants of the symmetric group, an important algebra isomorphic to the cohomology algebra of the flag variety.