Cluster parking functions II: $q,t$-dihedral sieving via diagonal coinvariants

Abstract: In a previous work, we defined the complex of cluster parking functions. On one side, they encode the type-refined enumeration of faces of the cluster complex, and on the other side, they have a reduced homology which is isomorphic to (ungraded) diagonal coinvariants. The goal of this work is to take into account the underlying dihedral symmetry. We thus have a product of a dihedral group and a symmetric group (there is a precise conjecture in the case of other finite Coxeter groups, but we focus on symmetric groups because of technicalities about diagonal coinvariants beyond this case). Under the action of the product group, the reduced homology of cluster parking functions is conjecturally isomorphic to diagonal coinvariants up to tensoring by a sign character of the dihedral group. This isomorphism can be reformulated as a dihedral sieving phenomenon. The main technical contribution is the definition of the dihedral automorphism group of cluster parking functions, and we discuss various features of the reduced homology character and its conjectural connection with diagonal coinvariants.

• The paper introduces a conjectural isomorphism between the reduced homology of cluster parking functions and the diagonal coinvariant module, revealing a q,t-dihedral sieving phenomenon under symmetric and dihedral actions.
• It employs explicit computations using rational Cherednik algebra modules and combinatorial dissections to connect parking functions with parabolic and dihedral symmetries.
• The findings provide new character identities and refined enumerative invariants that enhance our understanding of Coxeter group combinatorics and cluster complex representations.

$q,t$-Dihedral Sieving and Diagonal Coinvariants in Cluster Parking Functions

Overview and Context

The paper "Cluster parking functions II: $q,t$-dihedral sieving via diagonal coinvariants" (2607.04999) advances the study of combinatorial and representation-theoretic structures associated with cluster complexes, particularly in type $A$ and more generally for symmetric groups. The paper builds on prior work, extending the investigation into the interplay between cluster parking functions, dihedral symmetries, and diagonal coinvariant spaces. The core thematic focus is a conjectural isomorphism between the reduced homology of the cluster parking function complex, as a module over a product of symmetric and dihedral groups, and the diagonal coinvariants—up to a twist by a dihedral sign character.

Generalized Cluster Complexes and Parabolic Types

The cluster complex $\Gamma^{(m)}$ associated with a finite real reflection group $W$ and an integer $m \geq 1$ encapsulates rich Coxeter-theoretic and combinatorial properties. The combinatorial realization for type $A$ is via dissections of the $(mn+2)$-gon. Each face is assigned a parabolic type, reflecting the conjugacy class of associated parabolic subgroups. Enumeration refined by parabolic type is achieved through connections to parking functions—a coset space relevant in both noncrossing partition lattices and cluster theory. The dihedral symmetry of the cluster complex is generated by automorphisms $R$ and $S$ with relations $q,t$0, aligning naturally with the symmetries of regular polygons and their dissections.

Cluster Parking Functions and Dihedral Automorphisms

Cluster parking functions extend the paradigm by associating to each face $q,t$1 of the cluster complex a parking function $q,t$2, where $q,t$3 is the parabolic subgroup determined by $q,t$4. The homology of the resulting complex, $q,t$5, is ungraded and coincides with the diagonal coinvariant module, especially in type $q,t$6 for the case $q,t$7. The significant technical achievement of the paper is the formal extension of dihedral symmetry to cluster parking functions, maintaining the compatibility with both the action of the symmetric group and the complex's simplicial structure. This is described by explicit actions of $q,t$8 on $q,t$9, and their uniqueness is shown up to central elements of $A$0.

Diagonal Coinvariants: $A$1-Bigrading and Dihedral Action

Diagonal coinvariants, originally defined for the symmetric group $A$2 by Haiman, are constructed as the quotient of the polynomial ring in $A$3 variables by the ideal generated by constant-term-free $A$4-invariants. The ring is naturally bigraded by two variables $A$5, and carries a $A$6-action extended via the dihedral group $A$7 through specified automorphisms. This joint action leads to a refined character—the bigraded Hilbert series—that encodes rich combinatorial information, including $A$8-Catalan polynomials and their generalizations.

Notably, the paper conjectures an isomorphism (up to twist by a sign character) between the reduced homology of $A$9 under the action of $\Gamma^{(m)}$0 and the diagonal coinvariants: $\Gamma^{(m)}$1 where $\Gamma^{(m)}$2 is the character on homology, $\Gamma^{(m)}$3 the dihedral sign twist, and $\Gamma^{(m)}$4 the character of diagonal coinvariants. Furthermore, a $\Gamma^{(m)}$5-dihedral sieving phenomenon is posited, relating the evaluation of the bigraded Hilbert series at roots of unity (the eigenvalues of dihedral group elements) to homological invariants.

Numerical Results and Explicit Computations

The paper presents strong evidence for these conjectures in several cases:

• For rotations in the dihedral group, the evaluation of the diagonal coinvariant character at $\Gamma^{(m)}$6 simplifies to explicit formulae involving $\Gamma^{(m)}$7, matching the homology character of cluster parking functions and generalizing parking space enumeration in type $\Gamma^{(m)}$8 and $\Gamma^{(m)}$9.
• For reflections, connections to Euler and Springer numbers are established. Specialized evaluations (e.g., $W$0) relate to distinct combinatorial invariants, and computations in small rank Coxeter types (e.g., $W$1, $W$2, $W$3) confirm the conjectural identities.
• The paper provides explicit computational recipes using rational Cherednik algebra modules and twisted periods to construct the required coinvariant spaces, supporting the theoretical framework.

Contradictory and Bold Claims

The central claim is the conjectural module isomorphism between the reduced homology of $W$4 and the diagonal coinvariant ring up to sign twist, under the dihedral-symmetric group action. This is a strong assertion, especially in its generality for finite Coxeter groups beyond symmetric types, and is consistent with observed $W$5-dihedral sieving phenomena in explicit calculations. The extension to dihedral sieving goes beyond cyclic sieving and invokes deeper symmetry properties in cluster combinatorics.

Implications and Future Directions

The implications are both practical and theoretical:

• Representation Theory: The identification of homological structures in cluster complexes with diagonal coinvariants provides new pathways in understanding finite group actions, module decompositions, and symmetric functions.
• Enumerative Combinatorics: The $W$6-dihedral sieving phenomenon suggests new invariants and refined enumeration, generalizing Catalan-related counting and introducing Springer/Euler analogues in new contexts.
• Algebraic Geometry and Cherednik Algebras: The explicit connection to rational Cherednik algebra modules adds structure to the algebraic interpretation of parking spaces, pointing toward deeper geometric representation mechanisms.
• Extensions: While the conjectures are established in type $W$7 and dihedral types, the extension to higher diagonal coinvariants, other Coxeter types, and generalized cluster complexes remains an open area. Computational techniques and advances in symmetric functions theory will be essential.

Future research is expected to focus on:

• Explicit combinatorial models for cluster parking spaces in non-symmetric types.
• Further algebraic and homological testing of the conjecture in types $W$8, $W$9, and exceptional types.
• Intrinsic interpretation of sign twists and the structural origin of $m \geq 1$0-Springer numbers in broader contexts.
• Development of symmetric function theory to encode dihedral sieving for cluster complexes.

Conclusion

The paper delivers a technically rigorous extension of dihedral symmetry in cluster parking functions and conjectures their deep connection to diagonal coinvariants, leveraging combinatorial, homological, and algebraic approaches. Numerical and explicit computational evidence supports the $m \geq 1$1-dihedral sieving conjectures, advocating for a unified perspective on symmetries, parking spaces, and coinvariant algebra. The results provide new invariants and character identities, with broad implications for Coxeter group combinatorics and representation theory. Ongoing work will likely clarify the generality and robustness of these phenomena across types and in more sophisticated algebraic settings.