Euler-Poincaré characteristic and polynomial representations of Iwahori-Hecke algebras

The Hecke algebras of type A „ admit faithful representations by symmetrization operators acting on polynomial rings. These operators are related to the geometry of flag manifolds and in particular to a generalized Euler-Poincare characteristic denned by Hirzebruch. They provide g-idempotents, togetherwith a simple way to describe the irreducible representations of the Hecke algebra. The link with Kazhdan-Lusztig representations is discussed. We specially detail the case of hook representations, and as an application, we investigate the hamiltonian of a quantum spin chain with C/g(su(l/l)) symmetry.