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Journal Articles Seminaire Lotharingien de Combinatoire Year : 2005

Diagonal Temperley-Lieb Invariants and Harmonics

Abstract

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

Dates and versions

hal-00185498 , version 1 (06-11-2007)

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Jean-Christophe Aval, F. Bergeron, N. Bergeron. Diagonal Temperley-Lieb Invariants and Harmonics. Seminaire Lotharingien de Combinatoire, 2005, 54A, pp.B54Aq. ⟨hal-00185498⟩
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