Jordan Scheme in Algebraic Combinatorics

• Jordan scheme is a partition-based structure on finite sets whose adjacency algebra is closed under the Jordan product, generalizing association schemes.
• Proper Jordan schemes exist as non-symmetrisations of coherent configurations, demonstrated by minimal rank-5 examples and infinite families.
• The structure of their adjacency Jordan algebras reveals novel nonassociative components, linking combinatorial designs with RA-loops and alternative algebraic systems.

In algebraic combinatorics, a Jordan scheme is a homogeneous Jordan configuration, or equivalently a homogeneous coherent J-configuration, on a finite set whose adjacency algebra is closed under the Jordan product

$A*B:=\frac12(AB+BA)$

rather than under ordinary matrix multiplication. It generalizes association schemes in the same way that Jordan algebras generalize associative algebras: the combinatorial data are still encoded by a partition of $\Omega^2$, but the algebraic closure condition is weakened from associativity to Jordan closure. The subject was formalized by Peter Cameron, who asked whether there exist Jordan schemes that are not symmetrisations of coherent configurations; later work established the existence of such proper Jordan schemes and developed a structural theory around them (Muzychuk et al., 2019).

1. Formal definition and equivalent combinatorial description

Let $\Omega$ be a finite set and let $S$ be a partition of $\Omega^2$. The pair $(\Omega,S)$ is called a rainbow if $1_\Omega=\{(\omega,\omega):\omega\in\Omega\}$ belongs to $S$ and $S$ is closed under transposition: $S^t=\{s^t:s\in S\}=S$, where $\Omega^2$0. A Jordan scheme is a homogeneous coherent J-configuration, that is, a rainbow whose adjacency algebra

$\Omega^2$1

is closed under transpose, Schur-Hadamard product, and the Jordan product, and contains $\Omega^2$2 and $\Omega^2$3 (Muzychuk et al., 4 Sep 2025).

The elements of $\Omega^2$4 are the basic relations, or colors, and their adjacency matrices form a basis of the adjacency algebra. The term homogeneous means that the diagonal relation $\Omega^2$5 is present. A Jordan scheme is symmetric when every basic relation satisfies $\Omega^2$6, and regular when every basic relation $\Omega^2$7 has valency $\Omega^2$8 independent of $\Omega^2$9 (Muzychuk et al., 4 Sep 2025).

There is also a purely combinatorial formulation. If $\Omega$0 is a partition of $\Omega$1, then the Jordan configuration condition is that for any $\Omega$2, and for any $\Omega$3, $\Omega$4 in the same color class,

$\Omega$5

The corresponding intersection numbers are

$\Omega$6

where $\Omega$7 is the color class containing $\Omega$8. These numbers are the structural constants of the adjacency Jordan algebra (Muzychuk et al., 2019).

2. Relation to association schemes and coherent configurations

Association schemes and coherent configurations remain the basic comparison class. Association schemes are homogeneous coherent configurations whose adjacency algebras are closed under ordinary matrix multiplication. Jordan schemes replace this associative closure by closure under $\Omega$9, and this replacement is the source of both their flexibility and their nonassociative character (Muzychuk et al., 2019).

A major source of examples is symmetrisation. The symmetrisation of a coherent configuration, or of its adjacency algebra, yields a coherent J-algebra and hence a Jordan scheme. Such Jordan schemes are called non-proper or improper. A proper Jordan scheme is one that is not obtainable as the symmetrisation of any coherent configuration. In closure-theoretic terms, if $S$0 is a set of symmetric matrices, then $S$1 if and only if the corresponding coherent J-algebra is non-proper (Muzychuk et al., 2019).

This distinction resolves Cameron’s question in the affirmative. Proper Jordan schemes do exist, so the theory is not merely a reformulation of association-scheme theory with symmetrised multiplication. One immediate consequence is that Jordan closure and coherent closure are genuinely different operations, and Jordan schemes form a strictly larger class of algebraic-combinatorial objects than symmetrised coherent configurations (Muzychuk et al., 2019).

3. Proper Jordan schemes: existence, small examples, and infinite families

The first developments of the theory established both existence and minimality results for proper Jordan schemes. If $S$2 is a symmetric coherent J-configuration and is proper, then $S$3. Moreover, Jordan schemes of rank $S$4 are all improper, while a rank-$S$5 proper Jordan scheme has adjacency algebra isomorphic to

$S$6

(Muzychuk et al., 2019).

Small explicit examples were then exhibited. The first examples presented have orders $S$7, $S$8, and $S$9, and infinite classes of proper Jordan schemes of rank $\Omega^2$0 and larger were introduced (Klin et al., 2019). The order-$\Omega^2$1, rank-$\Omega^2$2 case also appears as the minimal case in a switching construction starting from non-commutative association schemes (Muzychuk et al., 2019).

Two systematic construction paradigms are already visible in the early theory. One is a rank-$\Omega^2$3 prolific construction outlined for schemes of order

$\Omega^2$4

and the other is a switching construction that begins with non-commutative association schemes and produces proper Jordan schemes after reorganizing the symmetric relations (Klin et al., 2019). In the later exposition of the theory, these appear as an infinite WFDF-based family of rank $\Omega^2$5 Jordan schemes and a second infinite family obtained by switching constructions from non-commutative association schemes (Muzychuk et al., 2019).

These results correct a common early expectation that Jordan schemes might all come from hidden associative structure. They show instead that properness is abundant enough to admit both sporadic and infinite families, while still being constrained enough that low-rank classification is possible (Klin et al., 2019).

4. Thin Jordan schemes and maximal rank-to-order ratio

A Jordan scheme is thin if every basic relation is thin, meaning that for all $\Omega^2$6, $\Omega^2$7; thus basic relations correspond to partial permutations. For a Jordan scheme $\Omega^2$8, the rank-to-order ratio is $\Omega^2$9. In the association-scheme case the maximum ratio is $(\Omega,S)$0, achieved only by thin schemes, but for Jordan schemes the non-regular bound is different: if $(\Omega,S)$1 is a non-regular Jordan scheme, then

$(\Omega,S)$2

with equality if and only if all basic relations are thin (Muzychuk et al., 4 Sep 2025).

The classification of thin Jordan schemes separates into non-regular and regular cases. Any non-regular thin Jordan scheme is, up to permutation, obtained from an abelian group $(\Omega,S)$3 through the block algebra

$(\Omega,S)$4

where $(\Omega,S)$5 is the group algebra with basis the permutation matrices associated to $(\Omega,S)$6. Conversely, any non-regular thin Jordan scheme arises from such a construction (Muzychuk et al., 4 Sep 2025).

For regular thin Jordan schemes, the basic relations are permutations containing the identity and closed under inverse. Fixing a basepoint $(\Omega,S)$7, one defines

$(\Omega,S)$8

With this operation, $(\Omega,S)$9 becomes a loop, and the classification theorem identifies the resulting structures exactly: regular thin Jordan schemes with maximal rank-to-order ratio correspond canonically to finite Ring Alternative Moufang loops (RA-loops). If such a Jordan scheme is symmetric, then it corresponds to an elementary abelian $1_\Omega=\{(\omega,\omega):\omega\in\Omega\}$0-group and is therefore an association scheme (Muzychuk et al., 4 Sep 2025).

Structure	Key condition	Corresponding object
Thin association scheme	Each basic relation is a permutation	Finite group
Non-regular thin Jordan scheme	Thin, not all relations are regular	“Doubled” abelian group construction
Regular thin Jordan scheme	Thin and regular, maximal rank-to-order ratio	Ring Alternative Moufang loop

The regular thin schemes arising from RA-loops are also autonomous: they cannot be obtained as an algebraic fusion of any coherent configuration (Muzychuk et al., 4 Sep 2025).

5. New proper Jordan schemes from quaternion and octonion data

A recent construction produces proper Jordan schemes from an elementary abelian $1_\Omega=\{(\omega,\omega):\omega\in\Omega\}$1-group $1_\Omega=\{(\omega,\omega):\omega\in\Omega\}$2 of rank $1_\Omega=\{(\omega,\omega):\omega\in\Omega\}$3 together with a $1_\Omega=\{(\omega,\omega):\omega\in\Omega\}$4-matrix $1_\Omega=\{(\omega,\omega):\omega\in\Omega\}$5 of order $1_\Omega=\{(\omega,\omega):\omega\in\Omega\}$6 satisfying

$1_\Omega=\{(\omega,\omega):\omega\in\Omega\}$7

From this data one forms $1_\Omega=\{(\omega,\omega):\omega\in\Omega\}$8 and defines a set of basic relations whose adjacency algebra is closed under the Jordan product by explicit calculations (Hanaki et al., 2 Sep 2025).

The admissible orders are sharply constrained. Using bilinear polynomials

$1_\Omega=\{(\omega,\omega):\omega\in\Omega\}$9

one obtains a composition identity

$S$0

and Hurwitz’s theorem implies that such matrices can exist only for orders $S$1, $S$2, or $S$3. Enumerating these cases yields three essentially distinct Jordan schemes: an improper scheme of order $S$4, and two new proper Jordan schemes of orders $S$5 and $S$6 (Hanaki et al., 2 Sep 2025).

The real adjacency Jordan algebras of these new schemes exhibit a previously unseen feature. For $S$7,

$S$8

and for $S$9,

$S$0

where the simple Jordan algebra $S$1 has product

$S$2

These are the first known Jordan schemes whose real adjacency Jordan algebras admit simple components of type $S$3, namely non-Hermitian type (Hanaki et al., 2 Sep 2025).

6. Adjacency Jordan algebras and structural significance

The algebra attached to a Jordan scheme is its adjacency Jordan algebra, the linear span of the basic adjacency matrices with product $S$4. In the recent constructions related to quaternion and octonion algebras, this real algebra is semisimple, formally real, and special, and it admits a decomposition through central idempotents

$S$5

(Hanaki et al., 2 Sep 2025).

The structure of the adjacency algebra often captures the combinatorial novelty of the scheme. At the minimal proper rank, the adjacency algebra is

$S$6

showing that properness already forces a genuinely nonassociative Jordan-algebraic component (Muzychuk et al., 2019). In the newer order-$S$7 and order-$S$8 constructions, the appearance of $S$9 and $S^t=\{s^t:s\in S\}=S$0 shows that proper Jordan schemes can realize simple components outside the Hermitian types previously seen in adjacency algebras (Hanaki et al., 2 Sep 2025).

Several misconceptions are therefore ruled out by the current theory. Not every Jordan scheme is a symmetrisation of a coherent configuration, because proper Jordan schemes exist in small orders and in infinite families (Klin et al., 2019). Not every maximal-ratio Jordan scheme is associative in disguise, because regular thin schemes can correspond to finite RA-loops rather than groups (Muzychuk et al., 4 Sep 2025). At the same time, symmetry remains restrictive: symmetric regular thin Jordan schemes collapse back to elementary abelian $S^t=\{s^t:s\in S\}=S$1-groups and hence to association schemes (Muzychuk et al., 4 Sep 2025).

Taken together, these results position Jordan schemes as a distinct class of algebraic-combinatorial structures. They interpolate between association schemes, coherent configurations, Jordan algebras, and loop-theoretic constructions, while retaining a precise matrix-algebraic formalism through transpose, Schur-Hadamard, and Jordan closure. The mature parts of the theory now include existence theorems, minimal-rank results, infinite families, thin-scheme classification, and explicit adjacency-algebra decompositions (Muzychuk et al., 2019).