Séminaire Lotharingien de Combinatoire, 91B.67 (2024), 12 pp.
Nantel Bergeron and Lucas Gagnon
Excedance quotients, Quasisymmetric Varieties, and Temperley-Lieb algebras
Abstract.
Let Rn
= Q[x1,x2,...,xn]
be the ring of polynomials in
n variables and consider the ideal
< QSymn+ > ⊆ Rn
generated by quasisymmetric
polynomials without constant term.
It was shown by Aval, Bergeron and Bergeron that
dim(Rn / < QSymn+ >)
= Cn the
nth Catalan number.
We explain here this phenomenon by defining a set of permutations
QSVn with the following properties: first,
QSVn is a basis of the Temperley-Lieb algebra
TLn(2), and second, when considering QSVn
as a collection of points in Qn, the top-degree
homogeneous component of the vanishing ideal
I(QSVn) is
< QSymn+ >. Our construction has a few byproducts
which are independently noteworthy.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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