FPP Conjecture: Bounds in Combinatorics & Representation

• FPP Conjecture is a sharp polyhedral bound that unifies aspects of combinatorics, representation theory, and statistical mechanics.
• It integrates refined enumeration techniques of fully packed loops with advanced methods such as Hodge filtration and parabolic induction.
• Its implications streamline the classification of unitary representations and enhance computational models in enumerative geometry and Banach space theory.

The FPP Conjecture is a sharp and widely studied polyhedral bound in several domains of mathematics and mathematical physics, encompassing combinatorics, representation theory, Banach space theory, and probability/statistical mechanics. Most commonly, "FPP" abbreviates either Fully Packed Loops or the Fundamental Parallelepiped bound in the context of unitary representation theory for real, complex, and p-adic reductive groups. In related research it appears as a constraint on refined enumerative combinatorics (e.g., link patterns and alternating sign matrices) or as a structural restriction on the infinitesimal character in the Langlands classification. Recent years have seen substantial progress: the FPP conjecture was recently proved for real reductive groups (Davis et al., 2 Nov 2024), complex simple Lie groups (Dong et al., 23 Jul 2024), and p-adic groups in pure rational form (assuming LLC) (Jiang et al., 22 Sep 2025); deep connections with combinatorial models (fully packed loops, alternating sign matrices) and fixed point theory in Banach spaces have also been revealed.

1. Combinatorial Formulation: Fully Packed Loops and Refined Enumeration

In combinatorics and statistical physics, the FPP conjecture arises as a refinement of the enumeration of Fully Packed Loop (FPL) configurations on square grids with specified link patterns. FPLs are subgraphs on an $n \times n$ square grid such that every vertex has degree two, and the selection of external edges follows rigid parity conditions, giving rise to "link patterns" (pairings of external edges).

• Key quantity: $A_X$, the number of FPLs with link pattern $X$.
• The Razumov–Stroganov correspondence encodes linear relations among these $A_X$ (Nadeau, 2011).
• Addition of nested arches to a pattern ($X \cup p$) produces $A_{X \cup p}$, which was shown to be a polynomial in $p$—explained via FPL configurations in a triangle (TFPLs).
• The refined enumeration collapses to formulae indexed by Dyck words, semistandard Young tableaux, and boundary word configurations:

$$A_{\pi}(m) = \sum_{\sigma, \tau \in D_n} \mathrm{SSYT}(\sigma, m-2n+1)\, t^{\pi}_{\sigma, \tau}$$

The paper of TFPLs enables a decomposition that proves the polynomiality of the refined numbers $A_X$, with degree governed by inversion numbers of the pattern and leading coefficients structured by hook product formulas. These linear relations (such as those conjectured by Thapper and subsequently proved) give recurrence relations distinct from the Razumov–Stroganov ones, imposing new combinatorial constraints and bridging to other models (alternating sign matrices, wheel polynomials) (Aigner, 2017).

2. Representation-Theoretic Formulation: Fundamental Parallelepiped Constraints

In the representation theory of real, complex, and p-adic reductive groups, the FPP conjecture asserts that the possible infinitesimal characters for unitary representations are constrained to a "fundamental parallelepiped" in weight space:

• For real reductive groups (Davis et al., 2 Nov 2024): If an irreducible unitary $(\mathfrak{g}, K)$-module has real infinitesimal character $\lambda$, then

$$\langle \lambda, \check{\alpha} \rangle \leq 1 \quad \text{for all simple coroots } \check{\alpha}$$

unless the representation is obtained by cohomological induction in the weakly good range.

• For complex simple Lie groups (Dong et al., 23 Jul 2024): The proof is via reduction to parabolic induction and explicit Hermitian form analysis on K-types ("cx-basic" cases). Non-unitarity outside the FPP region is established by signature computations on bottom layer K-types.
• For $p$-adic groups (Jiang et al., 22 Sep 2025): Using the (conjectural) Local Langlands Correspondence, the analogous statement is imposed on the exponent $\nu(\lambda_\phi)$ of the L-parameter: for a unitary irreducible smooth representation $\pi$, one must have

$$\langle \nu(\lambda_\phi), \alpha^\vee \rangle \leq 1 \quad \text{for all simple coroots } \alpha^\vee$$

The proof in pure rational form uses Vogan variety geometry and analytic deformations to exclude unitary exponents lying outside the FPP.

These boundaries yield a strong and uniform upper bound for the unitary dual, replacing more subtle or technical positivity conditions and enabling reduction steps in the unitary classification problem.

3. Methodological Innovations and Proof Strategies

The recent proof for real reductive groups (Davis et al., 2 Nov 2024) relies on advanced geometric and Hodge-theoretic techniques:

• Representation is realized as global sections of a $\mathcal{D}_\lambda$-module on the flag variety, with filtration by mixed Hodge modules.
• Global generation of each step in the Hodge filtration allows control over the underlying $\mathcal{O}_B$-module structure.
• The Adams–van Leeuwen–Trapa–Vogan unitarity criterion is formulated in terms of the Cartan involution acting on the associated graded pieces of the Hodge filtration—unitarity enforces a strict alternation of signs.
• Twisting by an appropriate line bundle and analysis of ample sections produces eigenvectors with incompatible signatures for the involution, violating the unitarity criterion if the infinitesimal character lies outside the FPP.

In the $p$-adic case (Jiang et al., 22 Sep 2025), the proof uses parabolic induction, analytic deformation via unbounded one-parameter families, and induction from suitably chosen Levi subgroups $M_{\leq 1}$ (constructed via geometric criteria on $\nu$). Compatibility with LLC is critical to carry over the constraints.

In the combinatorial setting (Nadeau, 2011), the methods revolve around the paper of TFPLs, Dyck word encodings, and bijective walks through configuration space—including the action of Wieland's rotation, decomposition into semistandard tableaux, and application of the Lindström–Gessel–Viennot lemma for determinantal formulas. This combinatorial structure mirrors and motivates algebraic recurrence constraints in representation theory.

4. Consequences for Unitary Duals, Classification, and Enumerative Geometry

The FPP conjecture delivers a robust control mechanism for both unitary representation theory and critical enumerative models:

• For reductive groups, it reduces classification of unitary duals to finite families of K-types with "small" infinitesimal characters.
• The constraint sharply restricts the possibility for unitary representations with "large" exponents, as shown explicitly by non-definiteness of Hermitian forms outside the FPP.
• In combinatorics/statistical physics, it enables polynomialness and explicit enumeration for fully packed loops and alternating sign matrices—filling the gap between probabilistic predictions and algebraic realizations.

In Banach space theory, "FPP" as fixed point property is likewise tightly correlated to geometric conditions (premonotonicity, unconditionality, Rosenthal-type properties, etc.), extending the theme of sharp bounding via underlying structure (Barroso, 2023).

5. Extensions, Generalizations, and Open Problems

Research inspired by the FPP conjecture proceeds along several axes:

• In higher-dimensional or non-Euclidean probabilistic models (e.g., FPP on trees, hyperbolic groups (Benjamini et al., 2013, Basu et al., 2019)), fluctuation bounds and coalescence phenomena echo the "tightness" predicted by FPP-type restrictions.
• Extension of FPP bounds to broader classes of groups (non-pure inner forms, possibly disconnected groups) remains an active direction.
• Computational applications—algorithms leveraging the FPP region for representation duals—are enabled by the sharpness and constructive nature of the constraint.
• Combinatorial implications for new classes of link patterns, loop models, and bijective correspondences continue to expand, with potential to reveal new integrable structures and symmetry relations.

6. Synthesis and Perspective

The FPP Conjecture integrates geometric, algebraic, and combinatorial viewpoints, imposing a uniform and sharply defined constraint in settings as disparate as unitary representation theory, loop models in planar graphs, Banach space geometry, and first passage percolation. The convergence of methods—from Hodge theory and D-module techniques in real groups to parabolic induction and analytic deformation in the p-adic context, and deep combinatorial bijections in planar models—underscores the principle that rich algebraic structure is intimately tied to geometric bounds and symmetry constraints. Whether restricting the support of infinitesimal characters or encoding recurrence relations for enumerative quantities, the FPP acts as a demarcation of viable models, facilitating both classification and constructive computation in modern mathematics.