Generalised Nice Set

• Generalised nice set is a combinatorial structure extending classical nice sets with an absorbing completion rule inspired by the Fano plane and octonions.
• The classification identifies 245 equivalence classes using PGL(3,2) collineation actions and algorithmic searches with combinatorial invariants.
• This framework underpins graded contractions of exceptional Lie algebras by encoding the support for fine Z₂³-gradings and prescribing algebraic limits.

A generalised nice set is a combinatorial structure introduced as an extension of classical nice sets on the Fano plane, with applications in the classification of graded contractions of exceptional complex Lie algebras endowed with $\mathbb{Z}_2^3$-gradings derived from octonion algebra. The concept is fundamentally rooted in the combinatorics of finite geometry—specifically, the incidence and symmetry structures of the Fano plane, and the algebraic structure induced by the octonions—serving as a unifying framework to describe the supports of gradings and contractions in these contexts (Draper et al., 15 Dec 2024).

1. Definition and Internal Structure

A generalised nice set is defined on an ambient finite set $X_0$, which expands the standard set $X$ of pairs (typically the “nice” part) associated with the Fano plane. It is a subset $T \subset X_0$ with two distinguished components:

• $T_{\mathrm{nx}} = T \cap X$, the classical nice part,
• $T - X$, the complementary part in $X_0 \setminus X$.

The essential property of (generalised) niceness is an absorbing “completion” rule derived from the geometry and algebra of the Fano plane and octonions. Specifically, given two elements $\{a,b\}$ and $\{a*b,c\}$ in $T$, the triple $P\{a, b, c\}$ (comprised of a line or triplet generated by $a, b, c$ using the octonion-based Fano plane multiplication *) must satisfy $P\{a, b, c\} \subset T$. This ensures that $T_{\mathrm{nx}}$ behaves as a classical nice set, while $T - X$ must be compatible with this absorption property in a sense made precise by the combinatorics of $X_0$.

Trivial yet non-obvious consequences of the definition, established by combinatorial closure and absorption arguments, include: if a “diagonal” element $\{i,i\}$ is in $T$, the corresponding “axis” element $\{0,i\}$ must also be included, and compatibility constraints interlink the structure of $T_{\mathrm{nx}}$ and $T - X$.

2. Methods of Classification

The classification of generalised nice sets is performed up to collineation, i.e., up to the action of the full collineation group $S^+(I)$ (identified with $\mathrm{PGL}(3,2)$ of order 168) of the Fano plane. The procedure involves:

• Disjoint case analysis by the relation between $T_{\mathrm{nx}}$ and $T - X$ (e.g., when $T \subset X$, $T \cap X = \emptyset$, or mixed cases).
• Further stratification by the cardinality $|T - X|$, with exhaustive tables documented for small sizes (such as the 13 classes for $T \subset X$, or for $|T - X| = k$ with $k$ from 1 to 8).
• Utilization of combinatorial invariants (“frequency sets,” “weights,” and “heights”) and necessary and sufficient absorption criteria (as formalized in key lemmas and theorems throughout the work).

Algorithmic and computer-assisted methods, including explicit group-theoretic searches and enumeration via invariants, are used for higher complexity cases (Section 7), ensuring completeness and independence of the classification.

In total, 245 equivalence classes of generalised nice sets (under collineation) are identified, each representing a fundamentally distinct combinatorial type relevant to graded contractions.

3. Role in Graded Contractions of Exceptional Lie Algebras

Generalised nice sets are directly linked to the construction of graded contractions of exceptional Lie algebras (such as $F_4$, $E_6$, $E_7$, $E_8$) with fine $\mathbb{Z}_2^3$-gradings. In this context:

• Each generalised nice set $T$ defines a support set for the grading, dictating which structure constants of the Lie bracket survive in the contraction.
• The set encodes the combinatorial pattern that specifies how to contract the algebra along the components corresponding to elements of $T$ under the grading.
• Two generalised nice sets related by a collineation give rise to isomorphic (i.e., equivalent under automorphisms) contracted Lie algebras, making the classification up to collineation essential for describing all distinct types of contractions attainable through this method.

4. Octonion Algebra and the Fano Plane Structure

The octonions $\mathbb{O}$, as a non-associative algebra, naturally produce a fine $\mathbb{Z}_2^3$-grading on the exceptional Lie algebras. The Fano plane serves as a geometric model for the multiplication rules of the basic units of the octonions, with each line in the plane corresponding to a triple of units whose product obeys a prescribed cyclic rule.

Within this framework:

• The generalised nice set’s absorption conditions reflect the closure properties of octonion multiplicative structure,
• The extension from $X$ to $X_0$ parallels the extension from standard “line-supported” gradings to more general support types,
• The combinatorial relations among elements of $T$ mirror the associator and commutator constraints arising in the algebraic bracket relations after contraction.

5. Combinatorial Features and Symmetry Considerations

The entire theory is governed by combinatorial methods, including:

• Enumeration of admissible subsets with respect to explicit geometric constraints,
• Application of the collineation group to identify equivalence classes and remove redundancy,
• Extensive use of invariants to streamline the isomorphism search between candidate sets.

All arguments and constructions in the classification are discrete and finite, independent of continuous or analytic structures, relying exclusively on the incidence geometry and symmetric group actions inherent in the Fano plane.

6. Applications and Broader Context

The comprehensive classification of generalised nice sets has significant algebraic consequences:

• It enables the systematic construction of all possible $\mathbb{Z}_2^3$-graded contractions of the exceptional complex Lie algebras with dimension at least 52,
• Provides a catalogue for the combinatorial data needed to prescribe such contractions (including solvable and nilpotent limits),
• Suggests directions for analogous classifications in other types of gradings or algebraic structures with strong symmetry and combinatorial underpinnings.

Since the core of the theory is combinatorial, further applications and generalizations are anticipated in settings where structure constants and algebraic operations can be controlled by discrete data associated with finite geometries and symmetry groups.