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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.
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https://hal.archives-ouvertes.fr/hal-00185498
Contributor : Jean-Christophe Aval <>
Submitted on : Tuesday, November 6, 2007 - 11:39:46 AM
Last modification on : Friday, March 27, 2020 - 2:55:56 AM

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

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