Hasse Diagram Summarization (HDS)

• Hasse Diagram Summarization (HDS) is a method that constructs Hasse diagrams to represent partial orders in complex data from statistics and supersymmetric gauge theory.
• It identifies hierarchical relationships through poset construction, enabling concise visualization and reproducible inference in experimental designs and geometric analysis.
• The framework automates degree-of-freedom computation and randomisation assignment, supporting robust analysis in both statistical and physical applications.

Hasse Diagram Summarization (HDS) provides a principled, algorithmic framework for extracting poset-structured summaries from complex combinatorial and geometric data, with key applications in experimental design (statistics) and symplectic singularities (supersymmetric gauge theory). At its core, HDS encodes the partial order among strata—derived from nesting, crossing, or algebraic stratification—into a Hasse diagram, enabling concise visualization and robust, reproducible statistical or geometric inference.

1. Formal Definition and Conceptual Foundations

HDS, in the classical mathematical-statistical setting, treats the set of “structural objects” in a design—factors and generalized factors generated by crossing (cartesian product) and nesting (refinement)—as a finite partially ordered set (poset). The partial order is typically that of refinement or coarsening: for factors $A$ and $B$, $A \le B$ if $A$ is nested in $B$, or $A$ is coarser than $B$. The resulting Hasse diagram depicts sources of variation and their hierarchical relationships. Annotating each node with its corresponding degrees of freedom via the subtraction method yields an ANOVA-like summary of the design’s structure (Michaelides et al., 8 Jul 2025).

In geometric representation theory and the theory of symplectic singularities, HDS encodes the closure relations among symplectic leaves or strata of a singular algebraic variety (e.g., the Higgs branch of a supersymmetric gauge theory) or among orbits in a representation space (Bourget et al., 2019). The stratification captures RG (renormalization group) flows, partial Higgsing, or other physical transitions.

2. Mathematical Construction and Algorithms

2.1 Poset Construction in Experimental Design

Given factor set $F=\{F_1, ..., F_k\}$, the layout structure (LS) is formed by all $F_i$ plus generalized factors $F_i \wedge F_j$ (full crossings) and nestings $B$0. The poset is ordered by nesting or refinement, with $B$1 denoting "A nested in B" and $B$2 denoting full crossing.

Degrees of freedom (DF) for each node $B$3 are allocated via the subtraction principle (Tjur):

$B$4

where the sum runs over immediate lower covers in the Hasse diagram (Michaelides et al., 8 Jul 2025).

Once randomisation is applied, one specifies randomisation arrows $B$5 between randomisation objects. Four algorithmic rules determine the restricted layout structure (RLS):

1. Include every $B$6 at the tail or head of a randomisation arrow.
2. Include all objects that randomisation-nest any object from rule 1.
3. Always include the overall mean and the observational unit factor.
4. If a generalised factor $B$7 is fixed and exists in LS, and each $B$8 is itself randomised, include it in RLS.

The resulting RLS is then visualized with directed arrows indicating randomisation, and mixed-model terms are directly read off from the diagram.

2.2 Quiver Subtraction in Symplectic Geometry

For Higgs branches and symplectic singularities, the poset of leaves is generated by the closure relations:

• Each node corresponds to a symplectic leaf (Coulomb branch of a magnetic quiver) (Lawrie et al., 2023, Bourget et al., 2019).
• Edges indicate transverse slices, usually quotient singularities or closures of minimal nilpotent orbits (ADE/Lie-type).

Algorithmically, construction proceeds as follows:

• Given a magnetic quiver $B$9, identify all balanced Dynkin subchains $A \le B$0.
• For each $A \le B$1, subtract its Dynkin labels from the quiver to produce a new quiver $A \le B$2 and record an edge labeled by the corresponding minimal orbit.
• Repeat until reaching trivial quivers.

For orthosymplectic quivers, symmetry mitosis is detected when the quiver branches into identical legs off a shared stem, requiring multiplicity (doubling) of certain edges in the Hasse diagram (Bennett et al., 15 Jan 2026).

3. Implementation and Software

HDS is implemented in the R package hassediagrams (Michaelides et al., 8 Jul 2025), providing key functions:

Function	Input / Role	Output / Visualization
hasselayout	Raw data.frame of factors, identifies LS, computes DFs, draws LS Hasse diagram	On-screen or PDF, optional LS table
itemlist	Data.frame of factors, builds full list of structural objects, returns rls object	To be edited for RLS construction
hasserls	rls object, user-edited randomisation assignment, random arrows	RLS Hasse diagram, model formula

The workflow involves:

1. Identification and encoding of factors and generalized factors.
2. Automated or manual assignment of randomisation objects and arrows.
3. Application of the HDS algorithm to generate and visualize LS and RLS diagrams.

Visualization is carried out with node/edge type annotation, degree-of-freedom labels, and optional mixed-model formula printout.

4. Illustrative Applications

4.1 Statistical Experimental Designs

Case studies include:

• Balanced incomplete block designs (BIBDs): LS includes mean, blocks, varieties, and their crosses; RLS restricts to those estimable under the chosen randomisation, suggesting directly the mixed-model formula (Michaelides et al., 8 Jul 2025).
• Multifactorial and split-plot designs: HDS recovers higher-order confoundings and identifies estimable terms in the post-randomisation model.

4.2 Symplectic Singularities and Supersymmetric Theories

In 6d $A \le B$3 little string theories (LSTs), HDS (via magnetic quiver subtraction) builds the Hasse diagram of the Higgs branch. The partial order encodes RG flows between higher and lower symplectic leaves, labeled by minimal orbits (e.g., $A \le B$4, $A \le B$5, etc.). Chains of such diagrams correspond to sequences of physical theories connected by partial Higgsing (Lawrie et al., 2023). Dimension and symmetry invariants (structure constants $A \le B$6, $A \le B$7, $A \le B$8) are monotonic along the poset, supporting physical interpretations such as candidate T-duality relations.

In orthosymplectic quivers with ON$A \le B$9-planes, the presence of symmetry mitosis requires combinatorial doubling of certain subtraction edges in the Hasse diagram, reflecting the generation of multiple factors of the associated global symmetry (Bennett et al., 15 Jan 2026).

5. Limitations, Generalizations, and Extensibility

Current HDS approaches in statistics are focused on multi-tier categorical designs; continuous covariates are not treated as sources in the poset. The user must supply or fine-tune randomisation objects and arrow assignments. Visualization may face platform-specific Unicode rendering limitations, and full automation of randomisation specification remains a potential extension (Michaelides et al., 8 Jul 2025).

In geometric and field-theoretic settings, the method presumes the availability of a magnetic quiver realizing the moduli space or its strata, and elementary slices are typically assumed to be closures of minimal nilpotent orbits or Kleinian singularities (Bourget et al., 2019, Lawrie et al., 2023). Extensions to arbitrary poset structures or to cases lacking a recognized quiver realization are plausible but not currently addressed.

A plausible implication is that HDS, as a mathematical paradigm and computational tool, is extensible to any finite stratification—statistical, geometric, or physical—admitting a computable partial order and suitable notion of DFs, slice dimension, or "capacity," and that its diagrammatic approach may be adapted for analysis outside of classical experimental design or supersymmetric geometry.

6. Significance and Connections

Hasse Diagram Summarization underpins rigorous model selection, ensuring that the terms included in statistical modeling or physical analysis match the information structure imposed by both design and randomisation. In statistics, HDS via hassediagrams guarantees reproducibility and transparency from raw data layout to final mixed-model structure (Michaelides et al., 8 Jul 2025). In high-energy physics and singularity theory, HDS reveals stratification patterns in moduli spaces, structures of RG flows, symmetry-breaking patterns, and dualities that are invisible to naive field-theoretic analysis (Lawrie et al., 2023, Bourget et al., 2019, Bennett et al., 15 Jan 2026).

By foregrounding the partial-order structure, annotating capacities, and systematically incorporating additional operations (such as symmetry mitosis), HDS serves as a unifying paradigm for summarizing stratified data in both discrete and geometric settings.