Quasisymmetric Divided Differences

• Quasisymmetric divided differences are linear operators on polynomial rings acting as non-symmetric analogues of classical Demazure divided differences, governed by Thompson-type monoid relations.
• They structure distinguished bases such as the monomial, fundamental, and forest bases, offering clear combinatorial and recursive methods for merging polynomial components.
• Their application in defining quasisymmetric coinvariant rings and modeling toric embeddings links fundamental concepts in combinatorics, algebraic geometry, and representation theory.

A quasisymmetric divided difference is a linear operator acting on the polynomial ring $\mathbb Q[x_1,\dots,x_n]$, designed as a non-symmetric analogue of the classical Demazure–BGG divided differences. These operators serve as the foundation for the theory of quasisymmetric Schubert polynomials, forest bases, and quasisymmetric coinvariant rings, and their algebra is controlled by Thompson-type monoid relations rather than the nil-Coxeter structure of the symmetric group. Quasisymmetric divided differences admit deep connections to combinatorics, algebraic geometry (especially the theory of toric and Schubert varieties), and representation theory of Hecke-type and 0-Hecke type algebras.

1. Formal Definition and Basic Properties

Let $\mathrm{Pol}_n = \mathbb Q[x_1,\dots,x_n]$ and let $\mathrm{QSym}_n \subset \mathrm{Pol}_n$ denote the ring of quasisymmetric polynomials. For $1\leq i\leq n$, define the Bergeron–Sottile specialization operator by

$$R_i : \mathrm{Pol}_n \to \mathrm{Pol}_{n-1}, \qquad R_i\bigl(f(x_1,\dots,x_n)\bigr) = f(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1}).$$

This operator inserts a zero in the $i$th slot and then shifts variables to the left. For $1\leq i\leq n-1$, the quasisymmetric divided difference is defined by

$$\partial_i^{\mathrm{qs}}(f) \coloneqq R_i\bigl(\partial_i^{\mathrm{sym}}f\bigr) = R_{i+1}\bigl(\partial_i^{\mathrm{sym}}f\bigr) = \frac{1}{x_i}(R_{i+1}f - R_if),$$

where $\partial_i^{\mathrm{sym}}$ is the Demazure (BGG) divided difference,

$$\partial_i^{\mathrm{sym}}(f) = \frac{f(x_1,\dots,x_n) - f(x_1,\dots,x_{i-1},x_{i+1},x_i,x_{i+2},\dots,x_n)}{x_i - x_{i+1}}.$$

Each $\mathrm{Pol}_n = \mathbb Q[x_1,\dots,x_n]$0 is $\mathrm{Pol}_n = \mathbb Q[x_1,\dots,x_n]$1-linear and of degree $\mathrm{Pol}_n = \mathbb Q[x_1,\dots,x_n]$2 (Nadeau et al., 2024, Nadeau et al., 2024).

2. Algebraic Structure and Thompson Monoid Relations

The operators $\mathrm{Pol}_n = \mathbb Q[x_1,\dots,x_n]$3 and the specializations $\mathrm{Pol}_n = \mathbb Q[x_1,\dots,x_n]$4 satisfy commutation rules that define an "augmented Thompson monoid," replacing the nil-Coxeter relations of classical divided differences. The two principal types of relations are as follows:

• Thompson–monoid commutation:

$\mathrm{Pol}_n = \mathbb Q[x_1,\dots,x_n]$5

• Mixed commutation (with specializations):

$\mathrm{Pol}_n = \mathbb Q[x_1,\dots,x_n]$6

Unlike the symmetric (nil-Hecke) case, there is no nilpotency: $\mathrm{Pol}_n = \mathbb Q[x_1,\dots,x_n]$7. Instead, the operators distinguish themselves by being faithful representations of the augmented Thompson monoid (Nadeau et al., 2024, Nadeau et al., 2024). The compositional structure is therefore governed by associativity laws for plane binary forests, with deep combinatorial significance.

3. Action on Distinguished Bases and Forest Recursions

$\mathrm{Pol}_n = \mathbb Q[x_1,\dots,x_n]$8 admits several algebraically significant bases:

• Monomial basis $\mathrm{Pol}_n = \mathbb Q[x_1,\dots,x_n]$9: For a composition $\mathrm{QSym}_n \subset \mathrm{Pol}_n$0,

$\mathrm{QSym}_n \subset \mathrm{Pol}_n$1

• Fundamental basis $\mathrm{QSym}_n \subset \mathrm{Pol}_n$2: For $\mathrm{QSym}_n \subset \mathrm{Pol}_n$3,

$\mathrm{QSym}_n \subset \mathrm{Pol}_n$4

• Forest (quasisymmetric Schubert) basis $\mathrm{QSym}_n \subset \mathrm{Pol}_n$5: Indexed by plane binary forests $\mathrm{QSym}_n \subset \mathrm{Pol}_n$6, with recursion

$\mathrm{QSym}_n \subset \mathrm{Pol}_n$7

where $\mathrm{QSym}_n \subset \mathrm{Pol}_n$8 is the forest obtained by trimming the $\mathrm{QSym}_n \subset \mathrm{Pol}_n$9th leaf and $1\leq i\leq n$0 is the quasi-descent set of $1\leq i\leq n$1 (Nadeau et al., 2024, Nadeau et al., 2024).

The forest basis reflects the Thompson-monoid algebra: any composition of divided differences $1\leq i\leq n$2 can be uniquely associated to a binary forest $1\leq i\leq n$3, so that $1\leq i\leq n$4 has well-defined action on $1\leq i\leq n$5 and captures recursive geometric and combinatorial interpretations.

4. Geometric and Coinvariant Interpretations

Let $1\leq i\leq n$6 denote the quasisymmetric coinvariant ring. The operators $1\leq i\leq n$7 characterize $1\leq i\leq n$8 as the joint kernel: $1\leq i\leq n$9 and descend to operators on $$R_i : \mathrm{Pol}_n \to \mathrm{Pol}_{n-1}, \qquad R_i\bigl(f(x_1,\dots,x_n)\bigr) = f(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1}).$$0 (Nadeau et al., 2024, Nadeau et al., 2024).

Geometrically, the push-forwards defined by $$R_i : \mathrm{Pol}_n \to \mathrm{Pol}_{n-1}, \qquad R_i\bigl(f(x_1,\dots,x_n)\bigr) = f(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1}).$$1 are realized by explicit toric $$R_i : \mathrm{Pol}_n \to \mathrm{Pol}_{n-1}, \qquad R_i\bigl(f(x_1,\dots,x_n)\bigr) = f(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1}).$$2-bundles and closed embeddings inside the flag variety $$R_i : \mathrm{Pol}_n \to \mathrm{Pol}_{n-1}, \qquad R_i\bigl(f(x_1,\dots,x_n)\bigr) = f(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1}).$$3. Families of quasisymmetric Schubert cycles $$R_i : \mathrm{Pol}_n \to \mathrm{Pol}_{n-1}, \qquad R_i\bigl(f(x_1,\dots,x_n)\bigr) = f(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1}).$$4, indexed by nested forests, satisfy

$$R_i : \mathrm{Pol}_n \to \mathrm{Pol}_{n-1}, \qquad R_i\bigl(f(x_1,\dots,x_n)\bigr) = f(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1}).$$5

for $$R_i : \mathrm{Pol}_n \to \mathrm{Pol}_{n-1}, \qquad R_i\bigl(f(x_1,\dots,x_n)\bigr) = f(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1}).$$6. The cohomology of the union of these toric subvarieties, a toric complex $$R_i : \mathrm{Pol}_n \to \mathrm{Pol}_{n-1}, \qquad R_i\bigl(f(x_1,\dots,x_n)\bigr) = f(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1}).$$7, injects into $$R_i : \mathrm{Pol}_n \to \mathrm{Pol}_{n-1}, \qquad R_i\bigl(f(x_1,\dots,x_n)\bigr) = f(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1}).$$8 and realizes the Poincaré pairing (Nadeau et al., 2024).

5. Combinatorics, Forests, and Polyhedral Geometry

Thompson-monoid relations precisely mirror forest combinatorics: elements correspond to plane binary forests, and faces of the associated toric cycles $$R_i : \mathrm{Pol}_n \to \mathrm{Pol}_{n-1}, \qquad R_i\bigl(f(x_1,\dots,x_n)\bigr) = f(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1}).$$9 form a subdivision of the permutahedron, with top-dimensional faces combinatorially cubical and indexed by maximal $i$0-bundle sequences (Nadeau et al., 2024). Each such face may also be viewed as a face of a Gelfand–Zetlin polytope or a nested-forest “cube” in $i$1 via explicit linear projections.

The moment polytopes of these toric cycles encode the geometry of quasisymmetric coinvariant quotients; the combinatorics of “trimming” and “merging” parts in the bases admits direct polyhedral interpretation and positive expansion theorems for the forest basis (Nadeau et al., 2024).

6. Comparison to Classical Divided Differences and Hivert-Type Operators

Unlike classical Demazure (BGG) operators, which satisfy nilCoxeter relations and quadratics $i$2, the quasisymmetric divided differences admit no such nilpotency. The braid relations are replaced with Thompson monoid relations: $i$3 which connect quasisymmetric divided differences to the non-Coxeter, forest-labeled combinatorics, and model the geometry of toric embeddings and quasisymmetric Schubert cycles (Nadeau et al., 2024, Nadeau et al., 2024).

In parallel, Hivert's quasisymmetric divided difference operators are defined via swaps that act only when exponents are zero, yielding fundamental bases such as the fundamental slide, fundamental particle, and $i$4-theoretic multifundamental families, arising in key, atom, and Schur polynomial analogues (Pierson, 15 Aug 2025, Hicks et al., 2024). However, the operators of Hivert-type differ essentially in their algebraic and combinatorial structure from the Thompson-monoid governed operators above. A plausible implication is that multiple distinct frameworks for "quasisymmetric divided differences" coexist, with the Thompson-monoid setting encoding stronger geometric and forest-theoretic features, while Hivert analogues admit direct connections to 0-Hecke actions and explicit polynomial bases.

7. Extensions and Further Developments

All aspects above extend to $i$5-colored quasisymmetric functions, whose divided difference theory uses $i$6-trimming (or multi-trimming) operators with relations governed by $i$7-Thompson monoids and forest bases indexed by $i$8-ary forests. These generalizations support further developments in the harmonic theory of quasisymmetric coinvariant rings, including the resolution of the Aval–Bergeron–Li conjecture, and give rise to explicit duality between forest polynomials and polytope volumes in the theory of harmonics (Nadeau et al., 2024).

Further research explores geometric and representation-theoretic interpretations, particularly $i$9-theoretic and Hecke-type analogues, and their connections to deformations of 0-Hecke actions, multifundamental and glide polynomials, and the modularity properties of Hessenberg representation theory (Pierson, 15 Aug 2025, Guay-Paquet, 8 Jul 2025).

---

Key references: (Nadeau et al., 2024, Nadeau et al., 2024, Hicks et al., 2024, Pierson, 15 Aug 2025, Guay-Paquet, 8 Jul 2025).