Quantum quasi-symmetric functions and Hecke algebras

Abstract

The algebra of quasi-symmetric functions is known to describe the characters of the Hecke algebra of type at v = 0. We present a quantization of this algebra, defined in terms of filtrations of induced representations of the 0-Hecke algebra. We show that this q-deformed algebra admits a simple realization in terms of quantum polynomials. For generic values of q, the algebra of quantum quasi-symmetric functions is isomorphic to the one of noncommutative symmetric functions. This gives rise to a one-parameter family of Hilbert space structures on the algebra of noncommutative symmetric functions, as well as to new interesting bases.