The excedance quotient of the Bruhat order, Quasisymmetric Varieties and Temperley-Lieb algebras

Abstract

Let $R_n=\mathbb{Q}[x_1,x_2,\dots ,x_n]$ be the ring of polynomial 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 ${\mathsf{TL}}_n(2)$, and second, when considering ${\mathrm{QSV}}_n$ as a collection of points in ${\mathbb{Q}}^n$, the top-degree homogeneous component of the vanishing ideal $\mathbf{I}({\mathrm{QSV}}_n)$ is $⟨{\mathrm{QSym}}_n^{+}⟩$. Our construction has a few byproducts which 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 ${\mathsf{TL}}_n(2)$.

23 pages. Final draft (all proofs are complete)

References in corpus (5)

• Clusters, Coxeter-sortable elements and noncrossing partitions
• Cambrian Lattices
• Combinatorial Hopf algebras and generalized Dehn-Sommerville relations
• Ideals of Quasi-Symmetric Functions and Super-Covariant Polynomials for S_n
• Catalan paths, Quasi-symmetric functions and Super-Harmonic Spaces