Séminaire Lotharingien de Combinatoire, B54Aq (2006), 19 pp.

Jean-Christophe Aval, François Bergeron and Nantel Bergeron

Diagonal Temperley-Lieb Invariants and Harmonics

Abstract. In the context of the ring Q[x,y], of polynomials in 2n variables x=x1,...,xn and y=y1,...,yn, we introduce the notion of diagonally quasi-symmetric polynomials. These, also called diagonal Temperley-Lieb invariants, make possible the further introduction of the space of diagonal Temperley-Lieb harmonics and diagonal Temperley-Lieb coinvariant space. We present new results and conjectures concerning these spaces, as well as the space obtained as the quotient of the ring of diagonal Temperley-Lieb invariants by the ideal generated by constant term free diagonally symmetric invariants. We also describe how the space of diagonal Temperley-Lieb invariants affords a natural graded Hopf algebra structure, for n going to infinity. We finally show how this last space and its graded dual Hopf algebra are related to the well known Hopf algebras of symmetric functions, quasi-symmetric functions and noncommutative symmetric functions.

Received: July 13, 2005. Accepted: October 31, 2006. Final Version: November 2, 2006.

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