Equivariant quasisymmetry and noncrossing partitions
Nantel Bergeron, Lucas Gagnon, Philippe Nadeau, Hunter Spink, Vasu Tewari
TL;DR
This work defines equivariant quasisymmetry for polynomials in two variable families, introducing the ring EQSym$_n$ and the double fundamental polynomials ${\mathfrak{F}_c}$ that specialize to ordinary fundamental quasisymmetric polynomials when the equivariant variables vanish. It then develops double forest polynomials ${\mathfrak{P}_F}(\mathbf{x};\mathbf{t})$, proves their existence and that they form a $\mathbb{Z}[\mathbf{t}]$-basis for $\mathbb{Z}[\mathbf{t}][\mathbf{x}]$, and establishes a subword (vine) model for their computation. A central theme is the evaluation of these polynomials at noncrossing partitions; a forest-to-noncrossing bijection ${\operatorname{ForToNC}}$ is constructed and shown to yield AJS–Billey-type formulas with a triangular, noncrossing structure that implies Graham-positivity of expansions and products. The paper introduces monoid structures (the Thompson/\star monoids and marked variants) to organize coefficient extraction via $\star$-compositions, culminating in a robust framework for positive combinatorics of equivariant quasisymmetric polynomials and their Schubert-forest expansions. The authors further outline a geometric program via a quasisymmetric flag variety to geometrize these invariants, indicating broad implications for toric and cohomological combinatorics beyond the symmetric case.
Abstract
We introduce a definition of ``equivariant quasisymmetry'' for polynomials in two sets of variables. Using this definition we define quasisymmetric generalizations of the theory of double Schur and double Schubert polynomials that we call double fundamental polynomials and double forest polynomials, where the subset of ``noncrossing partitions'' plays the role of $S_n$. In subsequent work we will show this combinatorics is governed by a new geometric construction we call the ``quasisymmetric flag variety'' which plays the same role for equivariant quasisymmetry as the usual flag variety plays in the classical story.