The excedance quotient of the Bruhat order, quasisymmetric varieties, and Temperley–Lieb algebras

Let $R_n=Q[x_1,x_2,\text{…},x_n]$ be the ring of polynomials in $n$ variables and consider the ideal $⟨{\mathrm{QSym}}_n^{+}⟩\subseteq R_n$ generated by quasisymmetric polynomials without constant term. It was shown by J. C. Aval, F. Bergeron, and N. Bergeron that $dim(R_n/⟨{\mathrm{QSym}}_n^{+}⟩)=C_n$ the $n$th Catalan number. In the present work, we explain this phenomenon by defining a set of permutations ${\mathrm{QSV}}_n$ with the following properties: first, ${\mathrm{QSV}}_n$ is a basis of the Temperley–Lieb algebra ${\mathrm{TL}}_n\left(2\right)$, and second, when considering ${\mathrm{QSV}}_n$ as a collection of points in $Q^n$, the top-degree homogeneous component of the vanishing ideal $I\left({\mathrm{QSV}}_n\right)$ is $⟨{\mathrm{QSym}}_n^{+}⟩$. Our construction has a few byproducts that are independently noteworthy. We define an equivalence relation $\sim$ on the symmetric group $S_n$ using weak excedances and show that its equivalence classes are naturally indexed by noncrossing partitions. Each equivalence class is an interval in the Bruhat order between an element of ${\mathrm{QSV}}_n$ and a 321-avoiding permutation. Furthermore, the Bruhat order induces a well-defined order on $S_n/\sim$. Finally, we show that any section of the quotient $S_n/\sim$ gives an (often novel) basis for ${\mathrm{TL}}_n\left(2\right)$.