Séminaire Lotharingien de Combinatoire, B54Aq (2006), 19 pp.
Jean-Christophe Aval, François Bergeron and Nantel Bergeron
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
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
Received: July 13, 2005.
Accepted: October 31, 2006.
Final Version: November 2, 2006.
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