Jordan Schemes in Combinatorial Algebra

• Jordan schemes are algebraic-combinatorial structures replacing associative closure with the Jordan product, establishing a novel framework in algebraic combinatorics.
• They are constructed from symmetric binary relations whose adjacency matrices span a finite-dimensional Jordan algebra, closed under both Jordan and Schur–Hadamard products.
• Proper Jordan schemes, which are not mere symmetrizations of association schemes, exhibit unique constructions at orders 15, 24, and 40 and extend to infinite families.

Jordan schemes are algebraic-combinatorial structures obtained by replacing the associative closure condition of association schemes with closure under the Jordan product $A*B=\frac12(AB+BA)$, together with closure under the Schur–Hadamard product. In their original formulation they arise from partitions of $\Omega^2$ into symmetric binary relations whose adjacency matrices span a finite-dimensional Jordan algebra containing the identity matrix and the all-ones matrix. They were introduced by Peter J. Cameron as a Jordan-theoretic analogue of association schemes, and the central early question was whether there exist proper Jordan schemes, meaning Jordan schemes that are not symmetrisations of association schemes or coherent configurations. That question was answered affirmatively by explicit constructions at orders $15$, $24$, and $40$, and the subject has since developed to include infinite families, extremal classification results for thin schemes, and new examples whose adjacency Jordan algebras contain spin-factor simple components (Klin et al., 2019, Muzychuk et al., 2019, Muzychuk et al., 4 Sep 2025, Hanaki et al., 2 Sep 2025).

1. Definition and formal framework

In the matrix-algebraic setting, one fixes a field $F$ of characteristic $0$ in the original essay, or more generally $\mathrm{char}(F)\neq 2$, and an order $n$. A coherent Jordan algebra is an $r$-dimensional subspace of $\Omega^2$0, typically inside $\Omega^2$1, that contains the identity matrix $\Omega^2$2 and the all-ones matrix $\Omega^2$3, and is closed under transposition, Schur–Hadamard multiplication, and the Jordan product. In the homogeneous case its standard basis consists of symmetric $\Omega^2$4–$\Omega^2$5 matrices $\Omega^2$6 with

$\Omega^2$7

and the Jordan multiplication has structure constants

$\Omega^2$8

The usual Jordan axioms, namely commutativity and the Jordan identity, are automatically satisfied by this matrix product (Klin et al., 2019).

Under the natural additional requirements used in the original development, this algebraic object is equivalent to a combinatorial partition of $\Omega^2$9 into symmetric binary relations. A Jordan scheme of order $15$0 and rank $15$1 may therefore be viewed as a pair $15$2, where $15$3, $15$4 is a partition of $15$5, $15$6, each $15$7 is symmetric, and the adjacency matrices $15$8 span a subspace closed under Schur and Jordan products. In this form, Jordan intersection parameters play the role that ordinary intersection numbers play in association schemes (Klin et al., 2019).

A later formalisation broadened the ambient language from symmetric partitions to rainbows, that is, partitions of $15$9 closed under transpose and with diagonal a union of basic relations. In that framework, a coherent Jordan configuration is a rainbow whose adjacency algebra is closed under the Jordan product, and a Jordan scheme is precisely the homogeneous case. The corresponding Jordan intersection numbers are

$24$0

so they are always non-negative and may be half-integral rather than integral. A practical recognition criterion is that, for a symmetric regular coloring with span $24$1, the coloring is a Jordan scheme if and only if $24$2 for all $24$3 (Muzychuk et al., 2019, Klin et al., 2019).

2. Relation to association schemes and the notion of properness

The foundational source of Jordan schemes is symmetrisation. If $24$4 is an association scheme or, more generally, a coherent configuration, then one may merge each relation $24$5 with its transpose $24$6. The resulting symmetric partition has adjacency span closed under the Jordan product, because $24$7 is the symmetrised associative multiplication. Such Jordan schemes are called non-proper or improper. A proper Jordan scheme is one that cannot be obtained in this way (Klin et al., 2019, Muzychuk et al., 2019).

This distinction is structurally significant. For symmetric matrix sets $24$8, one criterion states that the Jordan closure $24$9 equals the Weisfeiler–Leman coherent closure $40$0 if and only if $40$1 is non-proper. In the explicit infinite families, properness is often certified by proving that the coherent closure of the candidate Jordan scheme is inhomogeneous, whereas the symmetrisation of an association scheme is homogeneous. This provides an obstruction to being a symmetrisation rather than merely a computational test (Muzychuk et al., 2019, Klin et al., 2019).

The symmetric theory also exhibits a sharp low-rank constraint. A symmetric coherent Jordan configuration that is proper has rank at least $40$2. When equality holds, the configuration is homogeneous, and its adjacency Jordan algebra is isomorphic to

$40$3

This identifies the first non-associative proper case at the algebra level. By contrast, if the Jordan algebra were associative, then the configuration would collapse to a symmetric association scheme and hence be improper (Muzychuk et al., 2019).

Many regularity properties familiar from association schemes persist in modified form. In the original essay, each basic graph in a Jordan scheme is regular and, in the principal examples, walk-regular; unions of basic graphs are likewise walk-regular. Later work clarified that symmetric Jordan schemes are regular, whereas non-symmetric Jordan schemes need not be regular, a distinction that becomes important in the thin theory (Klin et al., 2019, Muzychuk et al., 4 Sep 2025).

3. First proper examples and landmark constructions

The first explicit proper Jordan schemes appeared at orders $40$4, $40$5, and $40$6. Their discovery answered Cameron’s existence question and established that properness is not a pathological exceptional phenomenon but a stable combinatorial possibility (Klin et al., 2019).

Order	Construction	Notable feature
$40$7	$40$8 by bridge switching from a rank-$40$9 non-commutative imprimitive association scheme	rank $F$0, $F$1
$F$2	$F$3 via switching from a scheme built from the Klein graph	rank $F$4, $F$5
$F$6	two proper Jordan schemes from Siamese color graphs	rank $F$7
$F$8	new proper schemes from elementary abelian $F$9-groups and $0$0-matrices	adjacency Jordan algebras with spin-factor components

The order-$0$1 example is especially important because it displays the geometric logic of the subject. One starts from a non-proper Jordan scheme $0$2 obtained by symmetrising a non-commutative association scheme of rank $0$3, then partitions the point set into an island of size $0$4 and a continent of size $0$5, constructs a pregraph with truncated tetrahedra and three bridge relations, and performs a bridge-switching operation. The resulting switched object is a proper rank-$0$6 Jordan scheme $0$7. The order-$0$8 example follows the same island–continent and switching philosophy, now using the Klein graph and a rank-$0$9 association scheme derived from the action of $\mathrm{char}(F)\neq 2$0 on $\mathrm{char}(F)\neq 2$1 perfect matchings of the Heawood graph (Klin et al., 2019).

A later development produced proper Jordan schemes of orders $\mathrm{char}(F)\neq 2$2 and $\mathrm{char}(F)\neq 2$3 from an elementary abelian $\mathrm{char}(F)\neq 2$4-group $\mathrm{char}(F)\neq 2$5 and a $\mathrm{char}(F)\neq 2$6-matrix $\mathrm{char}(F)\neq 2$7 satisfying a specified sign condition

$\mathrm{char}(F)\neq 2$8

The associated construction yields schemes on $\mathrm{char}(F)\neq 2$9 points with $n$0 basic relations. A Hurwitz-type composition-of-sums-of-squares argument forces $n$1, so only the orders $n$2, $n$3, and $n$4 occur; the order-$n$5 case is improper, while the orders $n$6 and $n$7 give proper Jordan schemes. The sign patterns are related to the composition algebras $n$8, $n$9, and $r$0 (Hanaki et al., 2 Sep 2025).

4. Infinite families, switching theory, and computational discovery

The first infinite proper families were obtained from non-commutative imprimitive association schemes built from $r$1 with $r$2. These schemes live on

$r$3

points and have three thin and three thick relations. Their symmetrisation gives a rank-$r$4 non-proper Jordan scheme $r$5; choosing one fiber as an island and switching two bridges while keeping the third thick relation fixed produces proper rank-$r$6 schemes $r$7. Properness is proved by showing that the coherent closure is inhomogeneous. Concrete instances include $r$8, giving $r$9 (Klin et al., 2019).

This construction admits a higher-rank generalisation with $\Omega^2$00 thick relations. In that setting the order is $\Omega^2$01, and bridge switching yields proper Jordan schemes of rank

$\Omega^2$02

The resulting schemes are pairwise combinatorially isomorphic within each cyclic orbit of the construction, showing that properness survives substantial variation in rank (Klin et al., 2019).

A second major source is the WFDF line of examples. The essay outlines a prolific rank-$\Omega^2$03 construction of order

$\Omega^2$04

based on three WFDF strongly regular graphs $\Omega^2$05 together with a Hoffman coloring. For $\Omega^2$06, corresponding to $\Omega^2$07, an extensive computer search found at least $\Omega^2$08 rank-$\Omega^2$09 Jordan schemes, all proper. Closely related later work produced a rank-$\Omega^2$10 family on

$\Omega^2$11

again built from three strongly regular graphs, with properness proved when $\Omega^2$12 is even (Klin et al., 2019, Muzychuk et al., 2019).

Computer experimentation was integral rather than auxiliary in this development. The authors used COCO and GAP, together with GRAPE and nauty, to compute $\Omega^2$13-orbits and centralizer algebras, enumerate mergings and fusions, determine automorphism groups, generate candidate color graphs, and test closure under Schur and Jordan products, including a Jordan stabilisation routine. Exhaustive or naive searches for $\Omega^2$14 and $\Omega^2$15 found no proper Jordan schemes; for $\Omega^2$16 the search remained ongoing, with conjectural nonexistence in those cases. The visual pregraph formalism, especially the island–continent model, was then abstracted into general switching theory (Klin et al., 2019).

5. Thin Jordan schemes and extremal rank-to-order phenomena

A substantial later advance was the classification of thin Jordan schemes. A basic relation is thin if every point has both in-degree and out-degree at most $\Omega^2$17 in that relation; a Jordan scheme is thin if every basic relation is thin. For non-regular Jordan schemes one has the sharp bound

$\Omega^2$18

where $\Omega^2$19 is the rank and $\Omega^2$20 the order, and equality holds if and only if all basic relations are thin. This shows that the Jordan setting permits a maximal rank-to-order ratio strictly larger than the group-theoretic value $\Omega^2$21 familiar from thin association schemes (Muzychuk et al., 4 Sep 2025).

The non-regular extremal case is completely classified. If a thin non-regular Jordan scheme has order $\Omega^2$22 and rank $\Omega^2$23, then it is permutation-conjugate to a block construction $\Omega^2$24 built from the adjacency algebra of a thin association scheme of an abelian group $\Omega^2$25 of order $\Omega^2$26. Concretely,

$\Omega^2$27

and every extremal thin non-regular Jordan scheme arises this way (Muzychuk et al., 4 Sep 2025).

The regular thin case leads outside group theory into nonassociative algebra. If $\Omega^2$28 is the set of basic permutations of a regular thin Jordan scheme and $\Omega^2$29 is defined from a base point by

$\Omega^2$30

then regular thin Jordan schemes are in one-to-one correspondence with Ring Alternative Moufang loops (RA-loops). Equivalently, the basic relations are the left translations of an RA-loop. When the loop is associative, the Jordan scheme is in fact a thin association scheme. This identifies RA-loops as the precise nonassociative replacement for groups in the regular thin Jordan setting (Muzychuk et al., 4 Sep 2025).

6. Adjacency Jordan algebras, later algebraic developments, and terminological scope

The adjacency algebra of a Jordan scheme is a special Jordan algebra because it is realized inside a matrix algebra under symmetrised multiplication. In the principal combinatorial theory it is also formally real and semisimple. This algebraic viewpoint is central to the analysis of idempotents, minimal ideals, and structural decompositions. In the rank-$\Omega^2$31 proper case, the real adjacency Jordan algebra is $\Omega^2$32, already indicating that proper Jordan schemes occupy a genuinely non-associative but still special part of Jordan theory (Muzychuk et al., 2019).

The order-$\Omega^2$33 and order-$\Omega^2$34 constructions pushed this analysis further. For the order-$\Omega^2$35 scheme $\Omega^2$36, the real adjacency Jordan algebra decomposes as

$\Omega^2$37

while for $\Omega^2$38,

$\Omega^2$39

where $\Omega^2$40 is the standard dot product and $\Omega^2$41 is the corresponding spin factor. These were identified as the first known proper Jordan schemes whose real adjacency Jordan algebras admit simple components of non-Hermitian type $\Omega^2$42 (Hanaki et al., 2 Sep 2025).

Several open problems remain central. The original and subsequent papers ask whether there exist proper primitive Jordan schemes, since the known proper constructions are imprimitive in the island–continent sense. Other open directions include classification for small orders, sharper algebraic criteria for properness, development of a spectral theory paralleling the Bose–Mesner and Euclidean Jordan settings, systematic study of automorphism groups, and extension of the known group- and WFDF-based construction mechanisms (Klin et al., 2019, Muzychuk et al., 2019).

The term Jordan schemes is not entirely stable across the wider literature. It has also been used for the Jordan property of automorphism groups of projective varieties, for geometric parameter spaces governing constant Jordan type in the representation theory of finite group schemes, and for a combined framework of emerging Jordan forms and dual Jordan quantum physics. This suggests a terminological ambiguity rather than a shared theory, and in algebraic combinatorics the standard meaning remains the partition-based Jordan analogue of association schemes described above (Meng et al., 2015, Pevtsova, 2014, Liu, 2024).