The goal of this paper is to give a new unified axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras. Building upon fundamental works by Young and Kazhdan-Lusztig, followed by Vershik and Ram, we propose a direct combinatorial construction, avoiding a priori use of external concepts (such as Young tableaux). This is carried out by a natural assumption on the representation matrices. For simply laced Coxeter groups, this assumption yields explicit simple matrices, generalizing the Young forms. For the symmetric groups the resulting representations are completely classified and include the irreducible ones. Analysis involves generalized descent classes and convexity (`a la Tits) within the Hasse diagram of the weak Bruhat poset.
|Title of host publication
|16th Conference in Formal Power Series and Algebraic Combinatorics
|Published - 2004