Diagonal Temperley-Lieb Invariants and Harmonics

Jean-Christophe Aval ; François Bergeron ; Nantel Bergeron

Séminaire lotharingien de combinatoire, 54A (2005-2007)

Résumé

In the context of the ring Q[x,y], of polynomials in 2n variables x=x₁,...,x_n and y=y₁,...,y_n, 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.