Quasisymmetric Flag Variety Insights

• Quasisymmetric flag variety is a geometric object that generalizes classical flag varieties using toric Bott manifolds and combinatorial structures like binary trees.
• It models equivariant quasisymmetric coinvariants by associating toric subvarieties to noncrossing partitions and employing recursive building operations.
• The framework bridges combinatorial algebra, geometry, and representation theory, offering new avenues for studying Schubert calculus and diagonal harmonics.

The quasisymmetric flag variety is a geometric object introduced to encode and realize the algebraic and combinatorial structure of quasisymmetric coinvariants, offering a direct analogue to the classical flag variety in the context of quasisymmetric function theory. The construction is based on toric geometry and deep combinatorial connections to noncrossing partitions, binary trees, and forest polynomials, serving as a foundational model for equivariant quasisymmetry and its cohomological invariants (Bergeron et al., 16 Aug 2025). The QFlₙ provides a rich framework that generalizes symmetric Schubert calculus and connects combinatorial algebra, geometry, and representation theory.

1. Structure and Construction

The quasisymmetric flag variety QFlₙ is defined as a union of subvarieties in the classical flag variety Flₙ:

$$\text{QFl}_n = \bigcup_{T \in \mathrm{Tree}_n} X(T) \subset \mathrm{Fl}_n.$$

Here, $\mathrm{Tree}_n$ denotes the set of planar binary trees with $n$ leaves, and each $X(T)$ is a toric Bott manifold associated to the tree $T$. The construction of $X(T)$ is inductive, applying certain "building operations"—denoted $\Psi_i^{-}, \Psi_i^{+},\mathcal{P}_i$—to toric varieties, starting from a point and iteratively constructing higher-dimensional Bott manifolds that correspond to the combinatorial structure of $T$. For a bicolored nested forest $F$, $X(F)$ is defined recursively via pattern maps and functorial operations. When $F$ corresponds to a top-dimensional forest, $X(F) = X(T)$ for some $T \in \mathrm{Tree}_n$; that is, each top cell is obtained by iteratively applying $\mathcal{P}_i$ operations to a point.

2. Toric Geometry and Noncrossing Partition Combinatorics

Each component $X(T)$ in QFlₙ is a smooth projective toric (Bott) manifold whose moment polytope is a cube, with the toric structure built via iterated $\mathbb{P}^1$-bundles. The vertices, faces, and fixed points of these cubes are parameterized by noncrossing partitions and bicolored nested forests. The torus $T_n$-fixed point set of QFlₙ is identified canonically with the set of noncrossing partitions $NC_n$ of $[n]$:

$\text{Fixed points of QFl}_n \simeq NC_n.$

Combinatorial stratifications of QFlₙ are governed by the Kreweras lattice, ordering noncrossing partitions by refinement. The faces of the moment polytope correspond to spreads of internal nodes in binary trees (or forests).

3. Equivariant Quasisymmetry and Coinvariant Cohomology

A central feature of QFlₙ is its role as a geometric model for equivariant quasisymmetric function theory. The $T_n$-equivariant cohomology ring is presented as:

$$H_{T_n}(\text{QFl}_n) \cong \mathbb{Z}[x_1, \ldots, x_n; t_1, \ldots, t_n] / (\mathrm{EQSym}_n^+),$$

where the ideal $\mathrm{EQSym}_n^+$ is generated by equivariantly quasisymmetric polynomials vanishing on the diagonal $x_i = t_i$. Upon specialization $t_i \to 0$, the nonequivariant cohomology ring becomes

$$H^{\bullet}(\text{QFl}_n) \cong \mathbb{Z}[x_1, \ldots, x_n] / \mathrm{QSymI}_n,$$

with $\mathrm{QSymI}_n$ generated by quasisymmetric polynomials without constant term. This realizes the ring of quasisymmetric coinvariants geometrically (Bergeron et al., 16 Aug 2025, Bergeron et al., 21 Apr 2025).

In parallel to symmetric Schubert calculus, the theory introduces divided difference operators—geometric analogues of Bott–Samelson operators—acting on the equivariant cohomology. The Bergeron–Sottile operators $R_i^-, R_i^+$ are linked to these divided differences, interpreted geometrically as pullback operations along the building maps.

4. Combinatorial Bases and Flow-up Structures

The cohomology rings admit combinatorial bases corresponding to double forest polynomials:

• Each double forest polynomial $P_F$ is constructed recursively, encoding the combinatorics of indexed forests (padded compositions) and their action under trimming operators.
• Evaluating double forest polynomials at torus-fixed points (given by noncrossing partitions) recovers upper-triangularity and Graham-positivity properties familiar from classical Schubert polynomials. The flow-up basis matches closures of torus orbits in homology.

A summary of the key structure is given in the following table:

Basis Type	Combinatorial Indexing	Evaluation at Fixed Points
Double forest polynomials	Indexed forests / compositions	Noncrossing partitions
Toric cell cycle classes	Binary trees / forests	Kreweras lattice refinements

These bases allow explicit computation and comparison with known coinvariant ring bases in the quasisymmetric regime.

5. Fixed Point Theory and GKM Graph Structure

GKM theory is deployed to compute equivariant cohomology via the graph cohomology of a GKM graph whose vertices are the noncrossing partitions and whose edges correspond to one-dimensional torus orbits (certain refinement moves in the Kreweras lattice). The equivariant localization ensures that the cohomology can be presented in terms of piecewise polynomial functions on NCₙ, with the combinatorics precisely tracking the geometry of the toric pieces $X(T)$.

This explicit combinatorial–geometric dictionary provides deep connections between the noncrossing partition lattice, the combinatorics of forest/trees, and the topology of the QFlₙ.

6. Applications and Implications

The construction of QFlₙ as a geometric model for quasisymmetric coinvariants has several significant consequences:

• Geometric realization of the ring $$\mathbb{Z}[x_1, \ldots, x_n] / (\text{quasisymmetric functions})$$, paralleling the classical flag variety's role for symmetric functions.
• The basis of double forest polynomials enables positivity and upper-triangularity results (Graham-positivity) in the coefficients of cohomological expansions.
• The toric complex's connection to noncrossing partitions provides new tools for exploring generalized permutahedra, matroid theories, cluster algebras, and free probability, implying that many structures traditionally studied algebraically or combinatorially are geometrically encoded in QFlₙ.
• The approach generalizes to equivariant settings, supporting paper of orbit harmonics and refined cohomological invariants associated to subgroups or other symmetry types.

A plausible implication is that the QFlₙ framework could be extended to model diagonal harmonics, parking function stratifications, and other phenomena in algebraic combinatorics that have previously lacked a direct geometric counterpart.

7. Relation to Previous Work and Future Directions

The quasisymmetric flag variety integrates previous developments involving quasisymmetric Schubert calculus (Pechenik et al., 2022), P-flag spaces and incidence stratifications (Bolognini et al., 2019), colored Eulerian quasisymmetric functions (Lin, 2013), and the topological models from loop spaces (Pechenik et al., 2022). The realization of the cohomology ring as the coinvariant ring of quasisymmetric polynomials closes a major gap in the theory, providing the foundational geometric object for further investigations in $K$-theoretic quasisymmetry, equivariant stratifications, and generalized Schubert calculus.

Potential future research directions include:

• Extension of the building operations and toric model to other combinatorial symmetry types and flag-like structures beyond binary trees and noncrossing partitions.
• Analysis of the singularities, orbit closures, and paving structures in QFlₙ for representation-theoretic and enumerative applications.
• Investigation of connections to cluster algebras via the poset structure arising from noncrossing partitions.
• Deployment of QFlₙ in the paper of diagonal harmonics, parking function modules, and harmonic coinvariants for additional combinatorial families.

In conclusion, the quasisymmetric flag variety QFlₙ is a toric subcomplex of the flag variety with fixed point set NCₙ and cohomology ring the quasisymmetric coinvariants, providing a geometric realization for the algebraic and combinatorial intricacies of quasisymmetric function theory (Bergeron et al., 16 Aug 2025).