A Unified Construction of Coxeter Group Representations – I

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, Kazhdan-Lusztig and Vershik, 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 (à la Tits) within the Hasse diagram of the weak Bruhat poset. Department of Mathematics and Statistics, Bar-Ilan University, Ramat-Gan 52900, Israel. Email: [email protected] Dipartimento di Matematica, Universitá di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy. Email: [email protected] Department of Mathematics and Statistics, Bar-Ilan University, Ramat-Gan 52900, Israel. Email: [email protected] Research of all authors was supported in part by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities; by EC’s IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272; and by internal research grants from Bar-Ilan University.