Quasisymmetric Coinvariants Overview

Updated 21 August 2025

Quasisymmetric coinvariants are a ring defined as the quotient of a polynomial algebra by the ideal of positive degree quasisymmetric functions, yielding a space whose dimension equals the Catalan numbers.
They incorporate refined descent statistics and are indexed by noncrossing partitions, planar binary trees, and Catalan objects, linking them to Temperley–Lieb algebras and flag varieties.
Recent developments extend these concepts via superspace variables, 0-Hecke algebra actions, and toric degenerations, offering new insights in algebraic combinatorics and geometric representation theory.

Quasisymmetric coinvariants constitute a central notion in algebraic combinatorics, combining the paper of coinvariant algebras, quasisymmetric functions, 0-Hecke and related algebraic actions, and connections to the geometry and topology of flag varieties and toric degenerations. Distinct from the classically symmetric coinvariant ring (the cohomology of the flag variety), the quasisymmetric coinvariant ring and its generalizations embody non-symmetric orbit harmonics, refined descent statistics, and finer combinatorial invariants linked to noncrossing partitions and more general posets. The topic unites advanced techniques in algebra, geometry, combinatorics, and representation theory.

1. Quasisymmetric Coinvariant Rings: Definitions and Classical Context

The classical coinvariant ring for the symmetric group $S_n$ is

$\mathrm{Coinv}_n = \mathbb{C}[x_1, \ldots, x_n]\,\big/\;\langle \text{Sym}^+_n\rangle$

where $\text{Sym}^+_n$ is the ideal generated by symmetric polynomials without constant term. The dimension is $n!$ and the ring forms the cohomology of the flag variety $\mathrm{Fl}_n$ , capturing the regular representation of $S_n$ .

The quasisymmetric coinvariant ring is the quotient

$QSCoinv_n = \mathbb{C}[x_1,\ldots,x_n]\,\big/\;\langle \mathrm{QSym}^+_n\rangle$

where $\mathrm{QSym}^+_n$ is the ideal generated by quasisymmetric polynomials of positive degree. Foundational work (Aval–Bergeron–Bergeron) establishes that $\dim QSCoinv_n = C_n$ , the $n$ th Catalan number, linking the graded structure and multiplicative basis to Catalan and noncrossing partition combinatorics (Bergeron et al., 2023).

The ideal $\langle \mathrm{QSym}^+_n\rangle$ is radical, and its quotient is characterized by the property that only a basis indexed by certain noncrossing combinatorial data survives, sharply reducing the dimension relative to the symmetric case.

2. Algebraic and Combinatorial Structure

Quasisymmetric coinvariant rings possess bases indexed by Catalan objects, such as noncrossing partitions or certain planar binary trees. In (Bergeron et al., 2023), an explicit basis $\mathrm{QSV}_n$ is constructed, indexed by noncrossing partitions, and shown to form, via a homomorphism, a basis for the Temperley–Lieb algebra $\mathsf{TL}_n(2)$ . The topology of the vanishing set of $\mathrm{QSV}_n$ is such that the leading homogeneous components of its vanishing ideal coincide with the ideal of positive-degree quasisymmetric functions: $\operatorname{gr}(\mathbf{I}(\mathrm{QSV}_n)) = \langle \mathrm{QSym}_n^+ \rangle,$ providing a geometric explanation for the coinvariant quotient.

Permutation relations and statistics are controlled by noncrossing partitions: each equivalence class in the symmetric group (via a weak excedance relation) is a Bruhat interval indexed uniquely by a noncrossing partition. The maximal and minimal elements in each class are, respectively, structural bases of the algebra (e.g., $321$-avoiding permutations and the $\mathrm{QSV}_n$ basis).

In the presence of superspace variables (i.e., with both bosonic and fermionic indeterminates), as in (Angarone et al., 27 Apr 2024), the superspace coinvariant ring is the quotient

$SR = K[x_1, \ldots, x_n] \otimes \Lambda[\theta_1, \ldots, \theta_n]\,/\,I,$

with $I$ the ideal generated by invariants. The explicit basis, confirming the Sagan–Swanson conjecture, consists of monomials $x_1^{a_1}\cdots x_n^{a_n}\theta_J$ for $J\subset [n]$ and $a_i < st(J)_i$ , with $st(J)$ the staircase sequence attached to $J$ . This basis is constructed via a recursion reflecting the anti-symmetry induced by the exterior variables and staircase growth conditions, with deep ties to the combinatorics underlying quasisymmetric invariants.

The action of the $0$-Hecke algebra $H_n(0)$ on the coinvariant algebra, via Demazure operators, refines the decomposition to descent classes (and hence to ribbon statistics) (Huang, 2012), and the regular representation structure reflects these finer combinatorial gradings.

3. Geometric and Topological Models: The Quasisymmetric Flag Variety

The "quasisymmetric flag variety" $QFl_n$ (Bergeron et al., 16 Aug 2025) is constructed as a toric subcomplex of the classical flag variety $\mathrm{Fl}_n$ , with an open cover by Bott towers (toric Richardson varieties $X(T)$ indexed by planar binary trees). The fixed point set of the torus action on $QFl_n$ is parametrized by noncrossing partitions, distinguishing it structurally from the full flag variety, whose torus fixed points correspond to permutations $S_n$ .

The cohomology ring of $QFl_n$ is isomorphic to the quasisymmetric coinvariant ring: $H^*(QFl_n) \cong \mathbb{C}[x_1,\ldots,x_n] / \langle f(x_1,\ldots,x_n) - f(0,\ldots,0): f\in \mathrm{QSym}_n\rangle.$ The $T_n$ -equivariant cohomology admits a "GKM" presentation: it is the subring of functions from noncrossing partitions $NC_n$ to $\mathbb{C}[t_1,\ldots, t_n]$ satisfying prescribed divisibility conditions along the edges of the noncrossing partition graph (an analogue of the Bruhat graph, but restricted to $NC_n$ ). The construction proceeds via equivariant pattern maps and projectivizations that recursively build up the Bott tower pieces, with the cube-like moment polytopes reflecting the Boolean lattice structure naturally embedded in $NC_n$ .

These geometric models produce natural duality between basis classes (such as "forest polynomials" and their dual volume polynomials), and subdivide the full flag variety into toric pieces with manageable combinatorics (see also (Nadeau et al., 16 Oct 2024, Nadeau et al., 3 Jun 2024)). For each toric piece, degrees against cohomology classes are computed via compositions of quasisymmetric analogues of Demazure and divided difference operators.

4. Operator Theory, Forest Polynomials, and the Thompson Monoid

A crucial innovation is the use of "quasisymmetric divided differences" (or trimming operators) as in (Nadeau et al., 3 Jun 2024, Nadeau et al., 16 Oct 2024). These operators, denoted typically $\tope{i}$, act by combinations of substitution and difference: $\tope{i} f = \frac{ f(x_1, \ldots, x_{i-1}, x_i, 0, x_{i+1},\ldots) - f(x_1,\ldots, x_{i-1}, 0, x_i, x_{i+1},\ldots)}{x_i}$ and satisfy commutation relations modeled by the Thompson monoid: $\tope{i} \tope{j} = \tope{j} \tope{i+1}, \quad i > j.$ Bases of the coinvariant ring are explicitly constructed as "forest polynomials" indexed by m-ary forests. These polynomials satisfy divided difference recursions analogous to the classical Schubert polynomials, but with combinatorics governed by forests instead of permutations: $\tope{i} \mathfrak{F}_F = \begin{cases} \mathfrak{F}_{F/i} & \text{if } i\in \mathrm{LT}(F)\ 0 & \text{otherwise}, \end{cases}$ where $F/i$ is the forest $F$ with its leftmost terminal node $i$ trimmed, and $\mathrm{LT}(F)$ is the left terminal set.

This theory naturally generalizes to colored, $m$ -quasisymmetric, and equivariant settings, with bases indexed by m-forests, colored trees, or double forests, as in recent work (Nadeau et al., 3 Jun 2024, Daugherty, 15 Dec 2024, Bergeron et al., 21 Apr 2025).

5. Connections to Noncrossing Partitions and Temperley–Lieb Algebras

A distinctive feature of quasisymmetric coinvariants is their enumeration and basis structure via noncrossing partitions. The dimension of the coinvariant space is the $n$ -th Catalan number $C_n$ because the quotient erases all quasisymmetric invariants except those indexed by these objects (Bergeron et al., 2023, Bergeron et al., 16 Aug 2025).

There exists a natural bijection between forests (which index the basis elements in the ring) and the set of noncrossing partitions; evaluation at suitable specializations is governed by this correspondence. The set $\mathrm{QSV}_n$ of permutations associated with noncrossing partitions forms both a basis for the (quasisymmetric) coordinate ring, and—under the surjection from $\mathbb{C}S_n$ —a basis for the Temperley–Lieb algebra $\mathsf{TL}_n(2)$ . These classes form intervals in Bruhat order (maximal element is the $\mathrm{QSV}_n$ permutation, minimal is the $321$-avoiding permutation corresponding to the partition).

Bruhat order on $S_n/\sim$ (weak excedance equivalence) descends to the Kreweras lattice on $NC_n$ , integrating the combinatorial, order-theoretic, and representation theoretic perspectives.

6. Hopf Algebras, Specializations, and Further Generalizations

Quasisymmetric coinvariant rings are deeply related to the Hopf algebra structure of $\mathrm{QSym}$ and its generalizations to colored and partially commutative variables (Daugherty, 15 Dec 2024). The duality between noncommutative symmetric functions ( $\mathrm{NSym}$ ) and quasisymmetric functions ( $\mathrm{QSym}$ ), and the appearance of their colored, commutative, and noncommutative versions, reflect invariant-coinvariant duality at the combinatorial and algebraic level.

Principal and stable principal specializations of quasisymmetric functions yield Hilbert functions and counts of monomials modulo various ideals, connecting the algebraic invariants to enumerative and Ehrhart-theoretic structures (White, 2016). Extensions to forbidden composition complexes, superspace coinvariants, and Solomon–Terao algebras via hyperplane arrangements broaden the applicability of the coinvariant concept beyond $S_n$ and its standard representations (Angarone et al., 27 Apr 2024).

Recent developments in equivariant quasisymmetry (Bergeron et al., 21 Apr 2025) define analogues of double Schur and Schubert polynomials as "double forest polynomials," employ noncrossing partition combinatorics, and pave the way for a geometric theory where the quasisymmetric flag variety plays a role analogous to the classical flag variety for symmetric theory.

7. Impact and Future Directions

Quasisymmetric coinvariants unify combinatorial, algebraic, and geometric approaches to non-symmetric representation theory and partition statistics, with canonical connections to Catalan objects, toric varieties, and the geometry of orbit harmonics. The emergence of the quasisymmetric flag variety as a geometric incarnation of the coinvariant quotient provides a new tool to paper orbit harmonics, positive coefficients (Graham positivity), and generalized Littlewood–Richardson coefficients in a non-symmetric setting (Bergeron et al., 16 Aug 2025).

Potential future directions include further development of colored and partially commutative coinvariant theories, $K$ -theoretic and equivariant analogues, and deeper paper of the connections to cluster algebras and polypositroid geometry via the combinatorics of noncrossing partitions and toric degenerations. The role of the Thompson monoid and novel monoidal structures in generating and controlling operator relationships in these contexts remains a topic of active investigation.

Key Relations and Structures Table:

Object / Notation	Description / Role	Combinatorial Indexing
$QSCoinv_n$	Quasisymmetric coinvariant ring	Catalan objects, $NC_n$ , trees
$\langle \mathrm{QSym}^+_n\rangle$	Ideal of quasisymmetric polynomials of positive degree	Leading to Catalan quotient
$\mathrm{QSV}_n$	Basis of coinvariants / $\mathsf{TL}_n(2)$	Noncrossing partitions
Forest Polynomials $\mathfrak{F}_F$	Basis of $QSCoinv_n$ via forests/Thompson monoid actions	Forests, terminal labelings
$QFl_n$	Quasisymmetric flag variety, toric subcomplex in $\mathrm{Fl}_n$	Trees (planar binary, bicolored)
$H^*(QFl_n)$	Cohomology ring equals $QSCoinv_n$	$NC_n$ , GKM-graph structure

The paper of quasisymmetric coinvariants thus provides a robust framework interweaving non-symmetric invariant theory, combinatorial algebra, and the geometry of flag varieties, broadening traditional Schubert calculus to the field governed by noncrossing partitions and quasisymmetric phenomena.