Jordan Schemes in Algebraic Combinatorics

• Jordan schemes are algebraic-combinatorial structures defined by replacing standard matrix multiplication with the symmetrized Jordan product, ensuring closure under transpose and Schur product.
• They extend association schemes by distinguishing proper schemes (not arising from symmetrizations) from improper ones, thereby refining classical coherent configuration theory.
• Explicit examples such as J15, J24, and infinite families illustrate their utility in analyzing algebraic symmetry, fiber decompositions, and new non-Hermitian components in adjacency algebras.

Jordan schemes are the Jordan-algebraic analogue of coherent configurations and association schemes. Their starting point is the replacement of ordinary matrix multiplication by the Jordan product

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

an idea going back to B.V. Shah (1959) and later adopted in the terminology of Peter Cameron. In the modern combinatorial formulation, a Jordan scheme is a homogeneous coherent Jordan configuration: a partition of $\Omega^2$ whose adjacency algebra is closed under transpose, Schur product, and Jordan multiplication, and whose diagonal is one basic relation (Muzychuk et al., 2019, Muzychuk et al., 4 Sep 2025). The subject developed around two questions that now organize the theory: how closely Jordan schemes parallel ordinary association schemes, and whether there exist proper Jordan schemes not obtainable as symmetrizations of coherent configurations.

1. Algebraic-combinatorial definition

A coherent Jordan algebra is a subspace $A\subseteq M_\Omega(F)$ that contains $I_\Omega$ and $J_\Omega$ and is closed under

${}^T,\quad \circ,\quad *,$

where $\circ$ denotes Schur-Hadamard product and $*$ denotes Jordan product. Equivalently, in the terminology of the thin-classification paper, if $\mathfrak S=(\Omega,S)$ is a partition of $\Omega^2$ into basic relations and

$\Omega^2$0

then $\Omega^2$1 is a coherent Jordan algebra when it is closed under Jordan product, transpose, and Schur product (Muzychuk et al., 2019, Muzychuk et al., 4 Sep 2025).

The combinatorial counterpart is a rainbow, that is, a partition of $\Omega^2$2 into basic relations. A Jordan configuration is obtained when the corresponding adjacency algebra is coherent Jordan. In the homogeneous case one has

$\Omega^2$3

and this homogeneous coherent Jordan configuration is called a Jordan scheme (Muzychuk et al., 4 Sep 2025).

The matrix and relation pictures are equivalent. A $\Omega^2$4-closed subspace has a unique $\Omega^2$5-matrix basis, and that basis corresponds to a partition of $\Omega^2$6. In relation-theoretic form, the Jordan condition is expressed by a modified intersection-number identity: for all $\Omega^2$7,

$\Omega^2$8

The associated structure constants are half-integral: $\Omega^2$9 Thus Jordan schemes preserve the coherent-configuration philosophy of structure constants, but with symmetrized multiplication in place of associative multiplication (Muzychuk et al., 2019).

A useful matrix criterion appears in the essay treatment: for a symmetric regular coloring $A\subseteq M_\Omega(F)$0 with adjacency matrices $A\subseteq M_\Omega(F)$1, the structure is a Jordan scheme if and only if

$A\subseteq M_\Omega(F)$2

This criterion packages closure under the Jordan product into a family of ordinary quadratic containment conditions (Klin et al., 2019).

2. Properness, symmetrization, and basic structural properties

Every symmetrization of an association scheme gives a Jordan scheme. More generally, every symmetrization of a coherent configuration gives a Jordan configuration. This produces a large supply of examples, but also motivates the central distinction between improper and proper Jordan schemes. A Jordan scheme is called improper if it comes from symmetrizing a coherent configuration, and proper if it does not (Muzychuk et al., 2019).

This distinction is sharpened algebraically by closure operations. For a set of matrices $A\subseteq M_\Omega(F)$3, one has the coherent closure $A\subseteq M_\Omega(F)$4, the smallest coherent algebra containing $A\subseteq M_\Omega(F)$5, and the Jordan closure $A\subseteq M_\Omega(F)$6, the smallest coherent Jordan algebra containing $A\subseteq M_\Omega(F)$7. Always,

$A\subseteq M_\Omega(F)$8

For symmetric inputs, equality holds exactly when the Jordan closure is non-proper. Properness can therefore be detected by strict containment,

$A\subseteq M_\Omega(F)$9

which became one of the standard methods in explicit constructions (Muzychuk et al., 2019).

Several structural facts distinguish the Jordan setting from ordinary coherent-configuration theory. Every Jordan configuration has a fiber decomposition

$I_\Omega$0

Each basic relation lies either between one pair of fibers or in a union of the two directions between two fibers, and relations between distinct fibers are bi-regular. At the same time, in symmetric Jordan configurations, homogeneous does not imply regular; the paper gives a rank-4 homogeneous but non-regular example. This is a genuine departure from the classical coherent-configuration pattern (Muzychuk et al., 2019).

A fundamental obstruction theorem shows that proper symmetric Jordan configurations cannot occur at very small rank. If $I_\Omega$1 is a proper symmetric Jordan configuration, then

$I_\Omega$2

In the extremal case $I_\Omega$3, the configuration is homogeneous and the corresponding Jordan algebra is

$I_\Omega$4

Accordingly, rank $I_\Omega$5 is the first possible location for proper Jordan schemes (Muzychuk et al., 2019).

3. Existence results, first examples, and infinite families

Peter Cameron asked in 2003 whether there exist Jordan schemes other than symmetrizations of coherent configurations. The answer is affirmative: proper Jordan schemes exist, and there are explicit infinite families (Muzychuk et al., 2019).

The first explicit proper examples were obtained by computer search and switching constructions. The essay presents $I_\Omega$6, $I_\Omega$7, and order-$I_\Omega$8 examples as the first small proper Jordan schemes (Klin et al., 2019).

Example	Order / rank	Salient data
$I_\Omega$9	$J_\Omega$0	automorphism group of order $J_\Omega$1, isomorphic to $J_\Omega$2; spread $J_\Omega$3; three copies of $J_\Omega$4
$J_\Omega$5	$J_\Omega$6	three copies of the Klein graph; $J_\Omega$7, order $J_\Omega$8
order-$J_\Omega$9 examples	${}^T,\quad \circ,\quad *,$0	valencies ${}^T,\quad \circ,\quad *,$1; automorphism group ${}^T,\quad \circ,\quad *,$2 of order ${}^T,\quad \circ,\quad *,$3

The smallest example, ${}^T,\quad \circ,\quad *,$4, has a particularly concrete description. It consists of the identity relation, a spread ${}^T,\quad \circ,\quad *,$5, and three copies of the unique antipodal distance-regular graph ${}^T,\quad \circ,\quad *,$6 of order ${}^T,\quad \circ,\quad *,$7 and valency ${}^T,\quad \circ,\quad *,$8. The graph ${}^T,\quad \circ,\quad *,$9 is the line graph of the Petersen graph,

$\circ$0

with intersection array

$\circ$1

The next example, $\circ$2, again has rank $\circ$3; its basic graphs are three isomorphic copies of the Klein graph, a distance-regular antipodal cover of $\circ$4 with intersection array

$\circ$5

Both examples arise from switching a non-proper companion $\circ$6 into a proper scheme $\circ$7 (Klin et al., 2019).

The 2019 paper then established two infinite constructions. The first is a rank-$\circ$8 family based on the Wallis–Fon-Der-Flaass construction of strongly regular graphs. With

$\circ$9

the scheme has one relation $*$0 given by “same fiber, different point” and three further symmetric relations $*$1 defined via affine hyperplanes, linear epimorphisms $*$2, and a binary operation $*$3 satisfying

$*$4

The set

$*$5

forms a Jordan scheme, and when $*$6 is even these examples are proper. The smallest case has order $*$7 and rank $*$8 (Muzychuk et al., 2019).

The second construction starts with a non-commutative association scheme

$*$9

whose relations satisfy the multiplication table

$\mathfrak S=(\Omega,S)$0

$\mathfrak S=(\Omega,S)$1

with arithmetic modulo $\mathfrak S=(\Omega,S)$2 and $\mathfrak S=(\Omega,S)$3. After a switching operation that splits the $\mathfrak S=(\Omega,S)$4 into within-fiber and between-fiber parts, one obtains a Jordan scheme whose properness is proved by exhibiting a matrix in the coherent closure that is not in the Jordan closure. This yields proper Jordan schemes of arbitrary rank (Muzychuk et al., 2019).

4. Thin Jordan schemes and extremal classification

The theory of thin Jordan schemes generalizes the classical correspondence between thin association schemes and groups. For a rainbow $\mathfrak S=(\Omega,S)$5, the order is $\mathfrak S=(\Omega,S)$6, the rank is $\mathfrak S=(\Omega,S)$7, and the key parameter is the ratio

$\mathfrak S=(\Omega,S)$8

In association schemes one always has $\mathfrak S=(\Omega,S)$9, with equality exactly in the thin case. Jordan schemes admit a different extremal behavior because regularity can fail (Muzychuk et al., 4 Sep 2025).

For non-regular Jordan schemes, the extremal bound is

$\Omega^2$0

Equality holds if and only if all basic relations are thin, meaning

$\Omega^2$1

for all $\Omega^2$2 and $\Omega^2$3. This yields the notion of non-regular thin Jordan schemes (Muzychuk et al., 4 Sep 2025).

The non-regular thin case is classified by a block-matrix construction. If $\Omega^2$4 is a commutative homogeneous coherent algebra, define

$\Omega^2$5

When $\Omega^2$6 is the adjacency algebra of a thin association scheme coming from an abelian group $\Omega^2$7, the notation is

$\Omega^2$8

Every non-regular thin Jordan scheme with adjacency algebra $\Omega^2$9 is, up to combinatorial isomorphism, of the form

$\Omega^2$00

for an abelian group $\Omega^2$01 and a permutation matrix $\Omega^2$02 (Muzychuk et al., 4 Sep 2025).

The regular thin case is governed by loop theory. If every basic relation is a permutation of $\Omega^2$03, then after fixing a basepoint $\Omega^2$04 one defines

$\Omega^2$05

This turns $\Omega^2$06 into a loop with neutral element $\Omega^2$07, and the Jordan-scheme identities force that loop to be alternative and, in fact, Moufang. The main correspondence is:

Class	Characterization	Classification
non-regular thin	$\Omega^2$08, all basic relations thin	comes from $\Omega^2$09 for an abelian group $\Omega^2$10
regular thin	every basic relation is a permutation	corresponds exactly to RA-loops
autonomous thin	not an algebraic fusion of a coherent configuration	occurs for nonassociative RA-loops

The central theorem states that the permutations $\Omega^2$11, with $\Omega^2$12, form a thin Jordan scheme if and only if $\Omega^2$13 is an RA-loop. Here an RA-loop is a loop $\Omega^2$14 such that its loop ring $\Omega^2$15 is alternative for every commutative associative ring $\Omega^2$16. Thus regular thin Jordan schemes are in one-to-one correspondence with RA-loops, while the non-regular thin branch is controlled by abelian groups (Muzychuk et al., 4 Sep 2025).

5. Adjacency Jordan algebras and new non-Hermitian components

If $\Omega^2$17 is the set of basis matrices of a Jordan scheme, its real adjacency Jordan algebra is

$\Omega^2$18

with Jordan product inherited from matrices. This algebra is always formally real, hence semisimple. Its simple summands belong to the classical list: $\Omega^2$19 Because adjacency Jordan algebras of Jordan schemes are special, the exceptional algebra $\Omega^2$20 cannot occur (Hanaki et al., 2 Sep 2025).

A new construction from 2025 starts with an elementary abelian $\Omega^2$21-group $\Omega^2$22 of rank $\Omega^2$23 and a $\Omega^2$24-matrix

$\Omega^2$25

satisfying

$\Omega^2$26

From $\Omega^2$27 one constructs a family of symmetric $\Omega^2$28-matrices whose span is closed under Jordan product. The construction is highly rigid: using Hurwitz’s theorem, the admissible matrix orders are forced to be

$\Omega^2$29

These correspond to the sign patterns of multiplication tables for $\Omega^2$30, $\Omega^2$31, and $\Omega^2$32 (Hanaki et al., 2 Sep 2025).

Up to combinatorial isomorphism, this construction yields exactly three Jordan schemes: one of order $\Omega^2$33, one of order $\Omega^2$34, and one of order $\Omega^2$35. The order-$\Omega^2$36 scheme is improper, while the order-$\Omega^2$37 and order-$\Omega^2$38 schemes are proper and are denoted

$\Omega^2$39

Their real adjacency Jordan algebras are

$\Omega^2$40

$\Omega^2$41

The summands $\Omega^2$42 are simple formally real Jordan algebras of non-Hermitian type, sometimes called spin factors. These are the first known examples whose real adjacency Jordan algebras admit simple components of this type (Hanaki et al., 2 Sep 2025).

This development changes the structural picture of adjacency Jordan algebras. Earlier work had already shown that the exceptional Jordan algebra does not occur in the combinatorial setting; the new examples show that non-Hermitian simple factors do occur, and do so in explicit proper Jordan schemes of orders $\Omega^2$43 and $\Omega^2$44 (Hanaki et al., 2 Sep 2025).

6. Terminology, scope, and adjacent Jordan-theoretic usages

In the combinatorial literature, the formal meaning of Jordan scheme is the homogeneous coherent Jordan configuration described above. The phrase also appears in broader Jordan-theoretic settings, and the meanings are not identical.

In ordered Jordan geometry, a partially ordered Jordan algebra $\Omega^2$45 gives rise to a Jordan geometry $\Omega^2$46 with a canonical $\Omega^2$47-invariant partial cyclic order whose intervals are affine or projective images of the symmetric cone $\Omega^2$48. That paper explicitly presents a research program toward “Jordan schemes,” bounded symmetric domains, and order-theoretic geometry (Bertram, 2017).

In the representation theory of finite group schemes, “Jordan schemes” are described as the geometric and cohomological framework refining support varieties by tracking the Jordan form of $\Omega^2$49-nilpotent operators along $\Omega^2$50-points, one-parameter subgroups, or elementary subalgebras. In that usage, the central objects are local Jordan type, constant Jordan type, non-maximal rank varieties, and vector bundles on parameter spaces such as $\Omega^2$51 and $\Omega^2$52 (Pevtsova, 2014).

In birational geometry over finite fields, the phrase is used informally in connection with the Jordan property of groups: a group $\Omega^2$53 is Jordan if every finite subgroup $\Omega^2$54 contains a normal abelian subgroup of uniformly bounded index, and $\Omega^2$55-Jordan is the characteristic-$\Omega^2$56 variant

$\Omega^2$57

That usage concerns subgroup structure rather than coherent Jordan algebras (Zaitsev, 25 May 2026).

These adjacent usages do not alter the formal combinatorial definition. They do, however, show that the phrase “Jordan schemes” now sits at an intersection of several Jordan-theoretic programs: algebraic combinatorics, ordered Jordan geometry, representation theory, and group-theoretic Jordan phenomena. Within the combinatorial theory itself, the decisive facts are now established: proper Jordan schemes exist, thin Jordan schemes admit a sharp classification, and adjacency Jordan algebras already realize nontrivial semisimple phenomena beyond symmetrized association schemes (Muzychuk et al., 2019, Muzychuk et al., 4 Sep 2025, Hanaki et al., 2 Sep 2025).