Equivariant Quasisymmetry

• Equivariant quasisymmetry is a framework that generalizes classical quasisymmetric polynomials by ensuring that inserting equivariant parameters adjacent to variables yields invariant forms.
• It underpins the development of double fundamental and double forest polynomials, providing positive bases that extend classical Schur–Weyl dualities and aid in computing equivariant cohomology.
• The theory manifests geometrically in the quasisymmetric flag variety, where toric subvarieties and noncrossing partitions reveal deep combinatorial and topological connections.

Equivariant quasisymmetry is an organizing principle in multiple branches of mathematics and mathematical physics, capturing the interplay between symmetry, combinatorics, geometry, and dynamical systems by encoding invariance or covariance properties under group actions, generalized for structures lacking full symmetry. The theory has distinct but interrelated developments in the context of polynomials and flag varieties, geometric combinatorics, and representation theory, particularly through recent formulations that exploit noncrossing partitions and toric geometry to generalize Schur–Weyl dualities and equivariant cohomology frameworks.

1. Definition of Equivariant Quasisymmetry and Motivations

Equivariant quasisymmetry for polynomials generalizes classical quasisymmetry, which concerns invariance of polynomial coefficients under pattern-preserving relabelings of indices. Classical quasisymmetric polynomials $f(x_1,\ldots,x_n)$ satisfy the property that the coefficient of any monomial $x_{i_1}^{a_1} \cdots x_{i_k}^{a_k}$, for an increasing index sequence, depends only on the sequence of exponents, not on the particular indices.

The equivariant analog, introduced in (Bergeron et al., 21 Apr 2025), considers polynomials in two (or more) sets of variables, $f(x_1,\ldots,x_n; t_1, t_2, \ldots)$, called "non-equivariant" and "equivariant" variables, respectively. Equivariant quasisymmetry is defined via the condition that insertion of an equivariant parameter $t_i$ immediately to the right or left of $x_i$ yields equal polynomials for every $i$:

$R_i^+ f = R_i^- f, \ \forall\ i,$

where

$$R_i^-f(x_1,\ldots,x_n; t_1, \ldots) = f(x_1,\ldots,x_{i-1}, t_i, x_i, x_{i+1},\ldots,x_n; \ldots),$$

$$R_i^+f(x_1,\ldots,x_n; t_1, \ldots) = f(x_1,\ldots,x_{i-1}, x_i, t_i, x_{i+1},\ldots,x_n; \ldots).$$

This condition encodes a kind of "adjacency symmetry" between equivariant and non-equivariant variables, interpolating between full symmetric and quasisymmetric invariance, and brings the properties of certain double polynomials (e.g., double Schur and Schubert polynomials) and their generalizations—namely, double fundamental and double forest polynomials—into a broader algebraic framework.

Upon specializing all $t_i$ to 0 (or suppressing equivariant parameters), this definition recovers standard quasisymmetry. The equivariant symmetry notion is motivated by the need to understand positive bases for equivariant cohomology rings, structure coefficients, and compatible isotypic decompositions in settings governed by group actions interpolating between full and partial symmetry (Bergeron et al., 21 Apr 2025).

2. Double Fundamental and Double Forest Polynomials

The formulation leads to two main families of polynomials:

• Double Fundamental Quasisymmetric Polynomials: These generalize the classical fundamental quasisymmetric polynomials. Double fundamentals are indexed by padded compositions—sequences of the form $(0^{n-\ell}, a_1,\ldots, a_\ell)$ with $a_i>0$. They provide a basis of the ring of equivariantly quasisymmetric polynomials. When set $t_i=0$, they reduce to the classical fundamentals.
• Double Forest Polynomials: These serve as positive (in the sense of Graham positivity) analogs of double Schubert polynomials for the new theory. Indexed by "indexed forests" (collections of planar trees with labels), their recursive definition uses a "trimming operation" $\{i\}$ that mimics the removal of leaves in a forest:

$\{i\} P_F(;) = \begin{cases} P_{F/i}(; \#_1^i), & \text{if $i$is a terminal label in$F$}, \ 0, & \text{otherwise}, \end{cases}$

with normalization $P_{\mathbf{1}}(;) = 1$ for the singleton forest. A subword (vine) model further undergirds their combinatorics.

These double polynomials form bases closed under multiplication by equivariantly quasisymmetric polynomials, generalizing well-known results for symmetric and Schubert functions. A central and ongoing open problem is to provide combinatorial formulas for structure coefficients, such as those arising in expansions of double Schubert polynomials into the double forest basis and in their products (Bergeron et al., 21 Apr 2025).

3. Noncrossing Partitions and Combinatorial Parametrizations

A defining theme is the replacement of the symmetric group $S_n$ by noncrossing partitions $\mathrm{NC}_n$ and their permutation analogs. Specifically, evaluation maps $$ev_\sigma f := f(t_{\sigma(1)}, t_{\sigma(2)}, \ldots; \ldots)$$, for $\sigma\in S_n$, reveal that for any equivariantly quasisymmetric $f$, the set of noncrossing permutations forms a fundamental class of evaluations where $ev_\sigma f$ is constant on equivalence classes generated by certain "compatible" elementary transpositions.

Furthermore, a canonical bijection (denoted ForToNC) exists between indexed forests and noncrossing partitions, which forms the backbone of the combinatorial structure of the theory. This bijection guides evaluation, expansion, and positivity properties, and enables the derivation of generalized AJS–Billey formulas for evaluation of the double forest polynomials (Bergeron et al., 21 Apr 2025).

The role of noncrossing partitions is not only to parametrize basis elements but also to provide a replacement for the Bruhat order in applications to (quasi)symmetric coinvariants and flag variety geometry.

4. The Quasisymmetric Flag Variety: Geometric and Cohomological Structure

The geometric scaffolding for equivariant quasisymmetry is the quasisymmetric flag variety, $QFl_n$, introduced and developed in (Bergeron et al., 16 Aug 2025). This is defined as

$$QFl_n := \bigcup_{T \in \mathrm{Tree}_n} X(T) \subset Fl_n,$$

where each $X(T)$ is a translated Richardson toric variety corresponding to a planar binary tree $T$ with $n$ leaves. This complex is a union of toric subvarieties in the ambient flag variety $Fl_n$.

Properties of $QFl_n$ with critical significance include:

Property	Description
Torus fixed points of $QFl_n$	In bijection with noncrossing partitions
Cohomology ring $H^\bullet(QFl_n)$	Isomorphic to the quasisymmetric coinvariants $\operatorname{qscoinv}_n$
Cohomology basis	Given by double forest polynomials
Moment polytope geometry	Given by polypositroids, not ordinary convex polytopes
Equivariant cohomology presentation	Achieved via GKM theory, with edge-labeled graphs reflecting noncrossing combinatorics

The GKM (Goresky–Kottwitz–MacPherson) presentation of the equivariant cohomology $H^\bullet_{T_n}(QFl_n)$ therefore employs equivariantly quasisymmetric polynomials and encodes the combinatorics of noncrossing partitions directly in the associated graph structure.

The connection between $QFl_n$, noncrossing partitions, and equivariant quasisymmetry organizes geometric, algebraic, and combinatorial phenomena into an integrative framework analogous to that of symmetric functions and the classical flag variety, but with noncrossing combinatorics replacing the full Weyl group action (Bergeron et al., 16 Aug 2025).

5. Significance and Broader Context

The theory links several research strands:

• It generalizes Schubert calculus, bringing Schubert positivity and module structure over polynomial rings into the field of quasisymmetric and equivariantly quasisymmetric functions.
• The interplay between noncrossing partitions, trees, and flag varieties provides new invariants and basis constructions, with a combinatorial model for the cohomology of $QFl_n$ paralleling (yet crucially different from) the classical coinvariant model for $Fl_n$.
• The GKM-theoretic perspective situates equivariant quasisymmetry within a topological context, potentially tying to orbit harmonics, representation theory (quasisymmetric coinvariants), and algebraic geometry (toric and Richardson varieties).

The geometric realization via $QFl_n$ also offers new perspectives on the resolutions and degenerations of flag varieties, moment polytope theory, and connections with positroid and matroid combinatorics.

6. Open Problems and Future Directions

Much of the combinatorics remains to be fully elucidated, including:

• Explicit Graham-positive formulas for structure coefficients in the double forest basis expansion and multiplicative structure of the quasisymmetric coinvariant ring (Bergeron et al., 21 Apr 2025).
• An "equivariant Monk's rule" or shuffle product for double fundamental quasisymmetric polynomials, extending classical rules in Schubert calculus.
• Refined geometric analysis of $QFl_n$: its satellites, degenerations, and intrinsic Plücker relations, and connections to other objects in algebraic geometry (e.g., positroid varieties, polypositroid stratifications) (Bergeron et al., 16 Aug 2025).
• Coherent extension of the theory to quantum, affine, and other generalized flag varieties, and to settings with more general group actions or quasisymmetric parameters.

Further development may yield new algebraic and topological invariants, positive combinatorial rules, and deepen the understanding of the universal role of noncrossing combinatorics in geometry and representation theory.

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In summary, equivariant quasisymmetry, through its precise algebraic, combinatorial, and geometric formulations, provides a novel framework unifying the paper of quasisymmetric functions, noncrossing partitions, and flag varieties, with a theory centered on the quasisymmetric flag variety $QFl_n$ and its associated GKM and coinvariant structures (Bergeron et al., 21 Apr 2025, Bergeron et al., 16 Aug 2025).