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Euler-Poincaré characteristic and polynomial representations of Iwahori-Hecke algebras. (English) Zbl 0835.05085
The Iwahori-Hecke algebra \(H_n\) associated to the symmetric group \(S_n\) has a faithful representation as an algebra of operators of the polynomial algebra in \(n\) variables such that the symmetric polynomials are treated as scalars under the action of \(H_n\). These symmetrizing operators are the Newton divided differences and their deformations. The simple operators satisfy the Yang-Baxter relations. The authors give several expressions of the operators corresponding to a maximal permutation and recover the generalized Euler-Poincaré characteristic defined by Hirzebruch in the geometry of flat manifolds. Restricting the action of the Hecke algebra to weight spaces, the authors also recover one of the usual descriptions of its representations. They also obtain \(q\)-idempotents and give a \(q\)-analogue of the Specht representations as orbits of products of \(q\)-Vandermonde functions. The authors study different constructions of irreducible representations corresponding to hook partitions and describe them in terms of Kazhdan-Lustig graphs. This interpretation is applied to the diagonalization of the Hamiltonian of a quantum spin chain with the quantum superalgebra \(U_q ({\mathfrak su} (1/1))\) as a symmetry algebra.
Reviewer: V.Drensky (Sofia)

MSC:
05E10 Combinatorial aspects of representation theory
20C30 Representations of finite symmetric groups
14M15 Grassmannians, Schubert varieties, flag manifolds
20G45 Applications of linear algebraic groups to the sciences
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