Polygon Equations and Their Hierarchies

• Polygon equations are a family of algebraic and combinatorial relations that generalize the pentagon equation by encoding operator consistency in tensor spaces.
• They leverage the structure of Tamari orders and dual relations to systematically construct hierarchies linking polygon and simplex equations with broad applications in mathematics and physics.
• Explicit constructions via matrix, set-theoretic, and analytic methods demonstrate their impact on state-sum invariants, quantum algebra, and topological field theory.

Polygon equations constitute a rich family of algebraic and combinatorial structures that generalize the classical pentagon equation, itself a cornerstone in higher category theory, mathematical physics, and the paper of integrability. They encode consistency conditions among operators or maps associated with the combinatorics of polygonal (associahedral or Tamari) orders and have deep connections to the algebraic and topological frameworks that underpin simplex equations such as the Yang–Baxter and higher simplex relations. Recent work unifies and extends these notions, explicitly constructing hierarchies of polygon and simplex equations and revealing precise mechanisms connecting solutions at each level, with applications spanning category theory, topological quantum field theory, and integrable systems.

1. Fundamental Definitions and Hierarchies

Polygon equations are parameterized by an integer $n \geq 3$ and are best understood as relations among families of linear operators $T_{I}$ acting on tensor powers of vector spaces, with the indices $I$ labeling specific tuples of tensor factors. A canonical example is the pentagon (5-gon) equation: $T_{12} T_{13} T_{23} = T_{23} T_{12}$ acting on $V^{\otimes 3}$, generalizing to higher $n$ by imposing consistency on compositions of operators corresponding to maximal chains in Tamari lattices $T(N, N-2)$ (Dimakis et al., 2014, Müller-Hoissen, 2023). For each $n$, the $n$-gon equation, potentially together with its dual, organizes the algebra of factor compositions so that different "resolutions" (orders of application) yield identical overall maps. These equations admit both set-theoretic and linear (matrix or tensor) realizations, and their paper is closely linked to the combinatorics of the Tamari and Bruhat posets.

Simplex equations, by contrast, generalize the Yang–Baxter (2-simplex) and tetrahedron (3-simplex) equations and are governed combinatorially by higher Bruhat orders. The relationship between these two families is rooted in the three-color (blue/red/green) decomposition of Bruhat orders, whereby the blue (respectively, red) suborder corresponds to the (standard) polygon (respectively, dual polygon) equations, and the green "mixed" part mediates compatibility, reconciling the full simplex equation as an overview of its polygon parts (Dimakis et al., 2014, Müller-Hoissen, 2023, Mihalache et al., 14 Oct 2025).

2. Combinatorial and Algebraic Framework

Let $N \geq 3$. In the formulation aligned with higher Bruhat/Tamari orders, the polygon equation is associated with the Tamari order $T(N,N-2)$ and the following data:

• For each $(N-1)$-element subset $K$ of $[N]$, a map $T_K$ acts on a Cartesian product of (usually tensor) factors, whose structure is defined in terms of "odd" and "even" packets $P_o(K), P_e(K)$, as extracted from a lexicographic ordering of subsets (Dimakis et al., 2014, Müller-Hoissen, 2023).
• The $n$-gon equation asserts that two different resolutions (explicit maximal chains) in the Tamari poset yield the same global operator, i.e.,

$$\prod_{i=1}^{k+1} T_{a_i} = \prod_{j=k}^{1} T_{b_j}$$

with multi-indices $a_i, b_j$ specified by the Tamari chain data.

The dual $n$-gon equation arises from the dual Tamari order, with maps $S_{J}$ defined in parallel; both standard and dual polygon equations enter the construction of solutions to simplex equations.

A crucial innovation demonstrated in recent work is the explicit formulation of a compatibility (mixed) relation between a solution $T$ of the $n$-gon equation and a solution $S$ of its dual (Mihalache et al., 14 Oct 2025, Dimakis et al., 2014, Müller-Hoissen, 2023). This mixed relation is written (for $n=2k+1$) in terms of alternating compositions of $T$ and $S$ along specific multi-indices (given by combinatorially defined matrices $A_{2k+1}$, etc.), as in: $$T_{d_{k+1}} S_{e_k} T_{d_k} \cdots S_{e_1} T_{d_1} = S_{f_1} T_{g_1} S_{f_2} \dots T_{g_k} S_{f_{k+1}}$$ where the indices $(d, e, f, g)$ are carefully constructed to capture the combinatorial content of the mixed order.

3. Construction of Higher Equations from Polygon Data

Polygon equations not only provide a framework for their own solutions but also serve as the stepping stone for higher-order equations. The main result in (Mihalache et al., 14 Oct 2025) and foreshadowed in (Dimakis et al., 2014, Müller-Hoissen, 2023) is that a "commutative" (i.e., compatible under an explicit intertwiner) pair of solutions of the $n$-gon and dual $n$-gon equations yields, via stacking/composition or tensor product, solutions to the $(n-1)$- and $(n-2)$-simplex equations.

For example, the construction proceeds as follows:

• Given $T$ and $S$ satisfying the polygon, dual polygon, and mixed relations, define

$$R^{(n-1)} = \sigma_{1,2} \sigma_{3,4} \cdots \sigma_{2k-1,2k} \circ T_{2,4,\dots,2k} S_{1,3,\dots,2k-1}$$

where the $\sigma_{i,j}$ are permutation (swap) operators, and the $T_J, S_K$ act on carefully selected subsets of tensor factors.

• This $R^{(n-1)}$ then satisfies the $(n-1)$-simplex (i.e., higher Yang–Baxter-like) equation.
• Similarly, higher ($n-2$) and ($n-3$)-simplex equations can be constructed, in principle, through iterated stacking and compatibility, with systematic extensions possible to arbitrary $n$.

This construction is made explicit for both odd and even $n$. For example, in the pentagon (n=5) case, stacking a pentagon solution and its dual produces a solution to the 4-simplex (tetrahedron) equation, generalizing the classic Kashaev–Sergeev construction and the framework of [Dimakis–Hüller-Hoissen].

4. Examples and Explicit Algebraic Realizations

Explicit algebraic realizations of polygon equations have been developed in multiple settings:

• Matrix and Tensor Solutions: Several works (Dimakis et al., 2020, Wan, 9 Jul 2024) provide matrix and Grassmannian–parameterized solutions to the $(2n+1)$-gon equations, expressing operator entries as rational functions or Plücker coordinates. The construction associates the combinatorics of Pachner moves and polyhedral triangulations with matrix operations, encoding the polygon equations as equalities between matrix products corresponding to different flip sequences in a triangulation.
• Set-Theoretic Examples: Piecewise-defined set-theoretic maps, as in (Müller-Hoissen, 2023), realize the polygon, dual polygon, and mixed equations at the purely combinatorial or categorical level.
• Analytic and Special Function Examples: In the pentagon (and lower) cases, particular solutions are expressed in terms of hypergeometric functions, Rogers dilogarithms, and correspond to key functional equations in analysis and special function theory.
• The general higher $n$-gon case features growing algebraic and combinatorial complexity, with the mixed relations formulated in terms of recursively constructed index matrices and operator compositions.

5. Connections to Category Theory, Geometry, and Physics

The pentagon equation is central to the coherence of monoidal categories and quasi-bialgebras, encoding associativity up to isomorphism. Higher polygon equations extend these coherence constraints, and their duals and mixed relations reflect deeper structures in higher category theory, such as higher associators and cocycles. The relation to simplex equations enables an overview between associative and Yang–Baxter structures, with direct relevance for:

• Topological Quantum Field Theory, via state-sum invariants invariant under Pachner moves (retriangulation of manifolds) (Dimakis et al., 2020, Wan, 9 Jul 2024).
• Integrable systems, where polygon equations generalize classical and quantum consistency conditions (e.g., generalizations of the tetrahedron and Yang–Baxter equations).
• Algebraic Geometry and Representation Theory, through the combinatorics of Tamari/associahedra and the algebraic encoding of polytopal flips and mutations, as well as their realization via Plücker coordinates on Grassmannians.

6. Implications, Generalizations, and Further Directions

The hierarchy of polygon and simplex equations, together with their explicit compatibility conditions, provides a blueprint for constructing larger families of integrable and topological structures. Immediate implications include:

• Systematic methods for producing solutions of arbitrarily high simplex equations from polygon data, with the mixed relation acting as the organizing principle.
• Clarified structural connections between polygon equations and their duals, via three-color decompositions of higher Bruhat orders, and the algebraic mechanism by which these decompose and reconstruct simplex equations.
• New algebraic constructions for state-sum invariants, quantum groups, and categorical associators, including parameterized and degenerate solutions.
• Extension of these approaches to symmetric monoidal categories and beyond, enabling category-theoretic generalizations and applications.

A plausible implication, although not yet fully realized, is that the systematic stacking and compatibility mechanism may yield new families of topological invariants and new classes of quantum algebraic structures, further enriching the landscape of categorified and higher-dimensional algebraic equations.

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In summary, polygon equations generalize associativity in a hierarchy that encodes combinatorial, algebraic, and categorical structures, with explicit constructions allowing for the derivation of higher simplex equations. Their paper reveals unified mechanisms underlying state-sum invariants, monoidal category coherence, and integrability, and continues to drive new connections across mathematics and mathematical physics (Dimakis et al., 2014, Müller-Hoissen, 2023, Mihalache et al., 14 Oct 2025).