Diagonal Temperley-lieb Invariants and Harmonics

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 quasisymmetric 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 ∞. 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.