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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
2*n* variables **x**=*x*_{1},...,*x*_{n} and
**y**=*y*_{1},...,*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.

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

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