Strong Formal Subdivisions in Combinatorics

• Strong formal subdivisions are combinatorial abstractions that generalize classical geometric subdivisions by enforcing strong surjectivity and parity-sum conditions in lower Eulerian posets.
• They enable the decomposition of complex invariants like the cd-index and local h-polynomials, offering recursive formulas and explicit computations in poset theory.
• This framework bridges combinatorics, geometry, and algebra by linking subdivision data with Kazhdan–Lusztig–Stanley invariants and equivariant Ehrhart theory.

A strong formal subdivision is a combinatorial abstraction of geometric subdivisions, generalizing classical polyhedral and fan subdivisions to the category of lower Eulerian posets. This concept provides a framework for decomposing combinatorial invariants of posets, such as the $cd$-index and local $h$-polynomials, and establishing canonical correspondences with Kazhdan–Lusztig–Stanley (KLS) invariants and equivariant Ehrhart theory. The theory of strong formal subdivisions has been developed to unify and extend existing subdivisions in the paper of poset invariants and to serve as a bridge to deep algebraic and topological invariants arising in combinatorics and geometry (Stapledon, 20 Nov 2025, Stapledon, 20 Nov 2025, Dornian et al., 2020).

1. Lower Eulerian Posets and the Definition of Strong Formal Subdivision

Let $X$ and $Y$ be finite lower Eulerian posets equipped with rank functions $\rho_X, \rho_Y$ (that is, for any element $z'$ covering $z$, $\rho_B(z') = \rho_B(z) + 1$, and every closed interval in $B$ contains the same number of even- and odd-ranked elements).

An order-preserving, rank-increasing, surjective map $\sigma: X \to Y$ is called a strong formal subdivision (SFS) if it satisfies:

• Strong Surjectivity: For every $x\in X$, $y\in Y$ with $\sigma(x) \leq y$, there exists $x'\geq x$ with $\rho_X(x') = \rho_Y(y)$ and $\sigma(x') = y$.
• Parity-sum Condition (Incidence Parity): For every $x\in X$, $y\in Y$ with $\sigma(x) \leq y$,

$$\sum_{\substack{x\le x'\in X \ \sigma(x')=y}} (-1)^{\rho_Y(y) - \rho_X(x')} = 1.$$

Equivalently,

$$\sum_{\substack{x\le x'\in X \ \sigma(x')\le y}} (-1)^{\rho_Y(y) - \rho_X(x')} = \begin{cases} 1 & \text{if } \sigma(x) = y, \ 0 & \text{if } \sigma(x) < y. \end{cases}$$

These conditions rigorously generalize the axioms satisfied by geometric subdivisions and ensure that compositions and restrictions of strong formal subdivisions themselves yield strong formal subdivisions (Stapledon, 20 Nov 2025, Dornian et al., 2020).

2. The Non-Hausdorff Mapping Cylinder and Canonical Bijection

Given any order-preserving map $\sigma: X \to Y$ between posets, the non-Hausdorff mapping cylinder $\Cyl(\sigma)$ is the poset on $X \sqcup Y$ with covering relations:

• $x < x'$ if $x, x' \in X$ and $x < x'$ in $X$,
• $y < y'$ if $y, y' \in Y$ and $y < y'$ in $Y$,
• $x < y$ if $\sigma(x) \leq y$ in $Y$.

The mapping cylinder construction provides a functorial and combinatorial gluing of $X$ and $Y$ via $\sigma$. This leads to the canonical bijection:

• Each strong formal subdivision $\sigma: X \to Y$ gives rise to $(\Gamma, \rho_\Gamma, q)$, where $\Gamma = \Cyl(\sigma)$, $q = \hat{0}_Y$, and the rank function $\rho_\Gamma$ extends those on $X$ and $Y$.
• Conversely, given a lower Eulerian poset $(\Gamma, \rho_\Gamma)$ and a join-admissible $q \neq \hat{0}_\Gamma$, one obtains $$\sigma: X = \Gamma \setminus \Gamma_{\ge q} \to Y = \Gamma_{\ge q}$$ via $x \mapsto x \vee q$.

This bijection establishes a structural correspondence between subdivisions and pairs $(\Gamma, q)$, with $\Gamma$ a lower Eulerian poset and $q$ a non-minimal join-admissible element (Stapledon, 20 Nov 2025, Stapledon, 20 Nov 2025).

3. $cd$-Index, Local $cd$-Index, and the Mixed $cd$-Index

Given an Eulerian poset $B$ of rank $n+1$, the $cd$-index $\Phi(B; c, d)$ is a homogeneous non-commutative polynomial in variables $c, d$ that encodes the flag enumeration of the chains in $B$. For near-Eulerian posets (certain subsets of Eulerian posets satisfying parity conditions), the local $cd$-index $\ell^\Phi(P)$ refines $\Phi$ by isolating the local contribution of a poset $P$ in a subdivision.

For a strong formal subdivision $\sigma: X \to Y$ with associated mapping cylinder $\Gamma = \Cyl(\sigma)$, the $cd$-index satisfies a recursive formula:

$$\Phi(\Gamma) = \ell^\Phi(X) + \frac{1}{2} \Big[ \Phi(\overline{\partial X}) c + \Phi(\overline{X_{\le \hat0_Y}}) c \Phi(Y) + \sum_{\hat0_Y < y < \hat1_Y} (\ell^\Phi(X_{\le y}) c + \Phi(\overline{X_{<y}}) d ) \Phi([y, \hat1_Y]) \Big],$$

with all terms of strictly smaller rank (Stapledon, 20 Nov 2025).

The mixed $cd$-index is a polynomial in noncommuting variables $c', d', c, d, e$ which encodes both the subdivision and the base poset:

$$\Omega_\phi(c', d'; c, d, e) = \sum_{x \in B} \ell^\Phi_{\Gamma_x}(c', d') \cdot \Phi_{[x, \hat1_B]}(c, d, e),$$

where $\Gamma_x$ is the fiber over $x$ (Dornian et al., 2020).

These indices give rise to mixed and local $h$-polynomials via canonical maps and capture decomposition properties of subdivisions.

4. Relations to Kazhdan–Lusztig–Stanley Theory

The local $h$-polynomials associated to strong formal subdivisions are tightly linked to the invariants in KLS theory. For lower Eulerian posets with suitable kernels $\kappa_\Gamma$ in their incidence algebras, the left and right KLS functions $g_\Gamma, f_\Gamma$ satisfy canonical involution identities:

$$f_B^{\mathrm{rev}} = \kappa_B f_B, \quad g_B^{\mathrm{rev}} = g_B \kappa_B,$$

and assemble to a symmetric $Z$-function.

In the context of a subdivision $\sigma: X \to Y$, the local $h$-polynomial $\ell_\sigma$ and its differences govern the restriction of KLS functions:

$$g_\Gamma|_{X/Y} = \Delta\ell_\sigma \cdot g_\Gamma,$$

$$Z_\Gamma|_{X/Y} = -Z_\Gamma|_X \Delta\widehat{\ell_\sigma} + (\Delta\ell_\sigma)^{\mathrm{rev}} Z_\Gamma|_Y,$$

and similar formulas for $f_\Gamma$. These identities provide deep combinatorial and representation-theoretic structure, including applications to the computation and interpretation of relative $g$-polynomials in terms of local $h$-polynomials (Stapledon, 20 Nov 2025).

5. Equivariant and Polyhedral Generalizations

If a finite group $W$ acts by order automorphisms preserving a $W$-invariant rank function, the incidence algebra and all associated invariants admit natural $W$-equivariant analogues. In this context:

• The equivariant kernel satisfies $\kappa^{-1} = \kappa^{\mathrm{rev}}$.
• All KLS-theoretic and $Z$-function results extend to the $W$-equivariant setting, with explicit invariants computed via evaluation on fixed-point posets $B^w$.
• For $W$-invariant fans or polyhedral subdivisions, there is a canonical equivariant kernel via determinants of induced affine actions, making the entire formalism applicable to equivariant Ehrhart theory.

When applied to lattice polytopes and their subdivisions, equivariant Ehrhart series and $h^*$-series can be explicitly described in terms of subdivisions and associated local KLS data. If a $W$-invariant unimodular triangulation exists, the equivariant $h^*$-polynomial coincides with the $h$-polynomial derived from the strong formal subdivision structure (Stapledon, 20 Nov 2025).

6. Decomposition Theorem and Explicit Examples

The decomposition theorem for the $cd$-index, extended to strong formal subdivisions, states:

$$\Phi_\Gamma(c,d) = \sum_{x \in B} \ell^\Phi_{\Gamma_x}(c,d) \Phi_{[x, \hat1_B]}(c,d),$$

breaking complex invariants into local contributions from fibers over elements $x$. This result generalizes and organizes classical decompositions (e.g., for barycentric subdivisions and explicit combinatorial polytopal subdivisions).

Explicit calculations for small examples confirm the framework. For instance, for the barycentric subdivision $S_2 \to B_2$ (edge subdivided at its midpoint), the mixed $cd$-index is $\Omega_{S_2 \to B_2} = c + d'e$. For $S_3 \to B_3$ (triangle to hexagon), the expansion involves several $c$, $d$, $c'$, $d'$ terms, reflecting the finer structure of the subdivision (Dornian et al., 2020).

7. Significance and Applications

The machinery of strong formal subdivisions provides a robust abstraction of geometric and combinatorial subdivision phenomena, enabling:

• Precise decomposition and recursion formulas for $cd$-indices and $h$-polynomials.
• Unified connections among poset invariants, KLS theory, representation theory, and Ehrhart theory.
• Novel approaches to local and relative invariants (e.g., the connection between relative $g$-polynomials and local $h$-polynomials).
• Equivariant generalizations necessary for applications in localization, invariants under group actions, and combinatorial representation theory.

The framework thus represents an overview of combinatorial, topological, and algebraic techniques, with further developments anticipated in poset-theoretic models of geometric representation theory and algebraic combinatorics (Stapledon, 20 Nov 2025, Stapledon, 20 Nov 2025, Dornian et al., 2020).