The Hecke algebras of type An 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_4(su(1/1)) symmetry.
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Gérard H. E. Duchamp, Daniel Krob, Alain Lascoux, Bernard Leclerc, Thomas Scharf, Jean-Yves Thibon, Euler-Poincaré Characteristic and Polynomial Representations of Iwahori-Hecke Algebras. Publ. Res. Inst. Math. Sci. 31 (1995), no. 2, pp. 179–201DOI 10.2977/PRIMS/1195164438