Euler–Poincaré Characteristic and Polynomial Representations of Iwahori–Hecke Algebras

Abstract

The Hecke algebras of type $A_n$ 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–Poincaré characteristic denned by Hirzebruch. They provide $q$-idempotents, together with 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 $U_q(\mathfrak{su}(1/1))$ symmetry.