2023/02/21 by Bergeron, Nantel, Gagnon, Lucas · 2 citations
#05E05 #14L24 #20C08 #33D80 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2302.10814
Let Rn=ℚ[x1,x2,…,xn] be the ring of polynomial in n variables and consider the ideal ⟨ QSymn+⟩⊆ Rn generated by quasisymmetric polynomials without constant term. It was shown by J.~C.~Aval, F.~Bergeron and N.~Bergeron that dim(Rn/⟨ QSymn+ ⟩)=Cn the nth Catalan number. In the present work, we explain 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. We define an equivalence relation ∼ on the symmetric group Sn 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 QSVn and a 321-avoiding permutation. Furthermore, the Bruhat order induces a well-defined order on Sn/ ∼. Finally, we show that any section of the quotient Sn/ ∼ gives an (often novel) basis for TLn(2).