---
title: Jordan Schemes in Algebraic Combinatorics
url: https://www.emergentmind.com/topics/jordan-schemes
type: topic
---

# Jordan Schemes in Algebraic Combinatorics

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 [1912.04551][2509.04068]. 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
\[
\mathbb F[S]=\langle A_s:s\in S\rangle,
\]
then \(\mathbb F[S]\) is a coherent Jordan algebra when it is closed under Jordan product, transpose, and Schur product [1912.04551][2509.04068].

The combinatorial counterpart is a rainbow, that is, a partition of \(\Omega^2\) into basic relations. A Jordan configuration is obtained when the corresponding adjacency algebra is coherent Jordan. In the homogeneous case one has
\[
1_\Omega\in S,
\]
and this homogeneous coherent Jordan configuration is called a Jordan scheme [2509.04068].

The matrix and relation pictures are equivalent. A \(\circ\)-closed subspace has a unique \(\{0,1\}\)-matrix basis, and that basis corresponds to a partition of \(\Omega^2\). In relation-theoretic form, the Jordan condition is expressed by a modified intersection-number identity: for all \(C,D\in\mathcal C\),
\[
C(a,b)=C(a',b') \Rightarrow |C(a)\cap D^T(b)|+|D(a)\cap C^T(b)| = |C(a')\cap D^T(b')|+|D(a')\cap C^T(b')|.
\]
The associated structure constants are half-integral:
\[
p_{C,D}^{F}=\frac12\Big(|C(a)\cap D^T(b)|+|D(a)\cap C^T(b)|\Big),\qquad F=C(a,b).
\]
Thus Jordan schemes preserve the coherent-configuration philosophy of structure constants, but with symmetrized multiplication in place of associative multiplication [1912.04551].

A useful matrix criterion appears in the essay treatment: for a symmetric regular coloring \({\mathfrak X}=(\Omega,\{R_0,\dots,R_{r-1}\})\) with adjacency matrices \(A_i\), the structure is a Jordan scheme if and only if
\[
(A_i+A_j)^2\in \operatorname{span}\{A_0,\dots,A_{r-1}\}
\quad\text{for all }i,j\ge 1.
\]
This criterion packages closure under the Jordan product into a family of ordinary quadratic containment conditions [1911.06160].

## 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 [1912.04551].

This distinction is sharpened algebraically by closure operations. For a set of matrices \(X\), one has the coherent closure \(\mathrm{WL}(X)\), the smallest coherent algebra containing \(X\), and the Jordan closure \(J(X)\), the smallest coherent Jordan algebra containing \(X\). Always,
\[
J(X)\subseteq \mathrm{WL}(X).
\]
For symmetric inputs, equality holds exactly when the Jordan closure is non-proper. Properness can therefore be detected by strict containment,
\[
J(X)\subsetneq \mathrm{WL}(X),
\]
which became one of the standard methods in explicit constructions [1912.04551].

Several structural facts distinguish the Jordan setting from ordinary coherent-configuration theory. Every Jordan configuration has a fiber decomposition
\[
\Omega=\Omega_1\cup\cdots\cup\Omega_f.
\]
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 [1912.04551].

A fundamental obstruction theorem shows that proper symmetric Jordan configurations cannot occur at very small rank. If \(\mathcal C\) is a proper symmetric Jordan configuration, then
\[
|\mathcal C|\ge 5.
\]
In the extremal case \( |\mathcal C|=5 \), the configuration is homogeneous and the corresponding Jordan algebra is
\[
\mathbb R+\mathbb R+\operatorname{Sym}_2(\mathbb R).
\]
Accordingly, rank \(5\) is the first possible location for proper Jordan schemes [1912.04551].

## 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 [1912.04551].

The first explicit proper examples were obtained by computer search and switching constructions. The essay presents \(J_{15}\), \(J_{24}\), and order-\(40\) examples as the first small proper Jordan schemes [1911.06160].

| Example | Order / rank | Salient data |
|---|---:|---|
| \(J_{15}\) | \(15 / 5\) | automorphism group of order \(12\), isomorphic to \(A_4\); spread \(5\circ K_3\); three copies of \(\Delta=L(\Pi)\) |
| \(J_{24}\) | \(24 / 5\) | three copies of the Klein graph; \(\operatorname{Aut}(\operatorname{Kle}_{24})\cong PGL(2,7)\), order \(336\) |
| order-\(40\) examples | \(40 / 7\) | valencies \(1,1,2,9,9,9,9\); automorphism group \((E_9:\mathbb Z_4)\times \mathbb Z_2\) of order \(72\) |

The smallest example, \(J_{15}\), has a particularly concrete description. It consists of the identity relation, a spread \(5\circ K_3\), and three copies of the unique antipodal distance-regular graph \(\Delta\) of order \(15\) and valency \(4\). The graph \(\Delta\) is the line graph of the Petersen graph,
\[
\Delta=L(\Pi),
\]
with intersection array
\[
(4,2,1;1,1,4).
\]
The next example, \(J_{24}\), again has rank \(5\); its basic graphs are three isomorphic copies of the Klein graph, a distance-regular antipodal cover of \(K_8\) with intersection array
\[
(7,4,1;1,2,7).
\]
Both examples arise from switching a non-proper companion \(NJ_n\) into a proper scheme \(J_n\) [1911.06160].

The 2019 paper then established two infinite constructions. The first is a rank-\(5\) family based on the Wallis–Fon-Der-Flaass construction of strongly regular graphs. With
\[
V=\mathbb Z_3^d,\qquad \Omega=V\times\{0,1,\dots,r\},\qquad r=3^d-1,
\]
the scheme has one relation \(S\) given by “same fiber, different point” and three further symmetric relations \(R_1,R_2,R_3\) defined via affine hyperplanes, linear epimorphisms \(T_i:V\to\mathbb Z_3\), and a binary operation \(\diamond\) satisfying
\[
a\diamond a=0,\qquad x\mapsto a\diamond x \text{ is a bijection.}
\]
The set
\[
\{I_\Omega,S,R_1,R_2,R_3\}
\]
forms a Jordan scheme, and when \(d\) is even these examples are proper. The smallest case has order \(15\) and rank \(5\) [1912.04551].

The second construction starts with a non-commutative association scheme
\[
\mathcal E=(\Omega,\mathcal R)
\]
whose relations satisfy the multiplication table
\[
C_iC_j=C_{i+j},\qquad C_iS_j=S_{i+j},\qquad S_iC_j=S_{i+j},
\]
\[
S_iS_j=nC_{i-j}+\frac{n-1}{m}(S_0+\cdots+S_{m-1}),
\]
with arithmetic modulo \(m\) and \(m\mid(n-1)\). After a switching operation that splits the \(S_i\) 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 [1912.04551].

## 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 \((\Omega,S)\), the order is \(|\Omega|\), the rank is \(|S|\), and the key parameter is the ratio
\[
\frac{|S|}{|\Omega|}.
\]
In association schemes one always has \(|S|\le |\Omega|\), with equality exactly in the thin case. Jordan schemes admit a different extremal behavior because regularity can fail [2509.04068].

For non-regular Jordan schemes, the extremal bound is
\[
\frac{|S|}{|\Omega|}\le \frac32.
\]
Equality holds if and only if all basic relations are thin, meaning
\[
|s\omega|\le 1,\qquad |s^t\omega|\le 1
\]
for all \(s\in S\) and \(\omega\in\Omega\). This yields the notion of non-regular thin Jordan schemes [2509.04068].

The non-regular thin case is classified by a block-matrix construction. If \(\mathcal C\le M_n(\mathbb F)\) is a commutative homogeneous coherent algebra, define
\[
\mathcal J(\mathcal C)=
\left\{
\begin{bmatrix}
A & B\\
C & A
\end{bmatrix}
: A,B,C\in\mathcal C
\right\}\subseteq M_{2n}(\mathbb F).
\]
When \(\mathcal C\) is the adjacency algebra of a thin association scheme coming from an abelian group \(G\), the notation is
\[
\mathcal J(G):=\mathcal J(\mathcal A(G)).
\]
Every non-regular thin Jordan scheme with adjacency algebra \(\mathcal A\) is, up to combinatorial isomorphism, of the form
\[
\mathcal A=P^{-1}\mathcal J(G)P
\]
for an abelian group \(G\) and a permutation matrix \(P\) [2509.04068].

The regular thin case is governed by loop theory. If every basic relation is a permutation of \(\Omega\), then after fixing a basepoint \(\omega_0\in\Omega\) one defines
\[
a\diamond b:=S(\omega_0,a(b(\omega_0))).
\]
This turns \(S\) into a loop with neutral element \(1_\Omega\), 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 | \(\frac{|S|}{|\Omega|}=\frac32\), all basic relations thin | comes from \(\mathcal J(G)\) for an abelian group \(G\) |
| 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 \(\{\ell_a:a\in S\}\), with \(\ell_a(x)=a\diamond x\), form a thin Jordan scheme if and only if \((S,\diamond)\) is an RA-loop. Here an RA-loop is a loop \(L\) such that its loop ring \(R[L]\) is alternative for every commutative associative ring \(R\). 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 [2509.04068].

## 5. Adjacency Jordan algebras and new non-Hermitian components

If \(S=\{A_0=I,A_1,\dots,A_d\}\) is the set of basis matrices of a Jordan scheme, its real adjacency Jordan algebra is
\[
RS:=\bigoplus_{i=0}^d \mathbb RA_i,
\]
with Jordan product inherited from matrices. This algebra is always formally real, hence semisimple. Its simple summands belong to the classical list:
\[
\mathbb R,\qquad \mathbb R\oplus_f V,\qquad H_n(\mathcal D)\ (\mathcal D=\mathbb R,\mathbb C,\mathbb H),\qquad H_3(\mathbb O).
\]
Because adjacency Jordan algebras of Jordan schemes are special, the exceptional algebra \(H_3(\mathbb O)\) cannot occur [2509.01865].

A new construction from 2025 starts with an elementary abelian \(2\)-group \(G\) of rank \(n\) and a \(\{1,-1\}\)-matrix
\[
M=(M_{x,y})_{x,y\in G}
\]
satisfying
\[
M_{x,xz}\,M_{x,yz}\,M_{y,xz}\,M_{y,yz}=-1
\qquad
\text{for all }x,y,z\in G\text{ with }x\neq y.
\]
From \(M\) one constructs a family of symmetric \(\{0,1\}\)-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
\[
|G|=2,4,\text{ or }8.
\]
These correspond to the sign patterns of multiplication tables for \(\mathbb C\), \(\mathbb H\), and \(\mathbb O\) [2509.01865].

Up to combinatorial isomorphism, this construction yields exactly three Jordan schemes: one of order \(8\), one of order \(16\), and one of order \(32\). The order-\(8\) scheme is improper, while the order-\(16\) and order-\(32\) schemes are proper and are denoted
\[
J_{16},\qquad J_{32}.
\]
Their real adjacency Jordan algebras are
\[
RS(J_{16})\cong 8\mathbb R\oplus(\mathbb R\oplus_f\mathbb R^4),
\]
\[
RS(J_{32})\cong 16\mathbb R\oplus(\mathbb R\oplus_f\mathbb R^8).
\]
The summands \(\mathbb R\oplus_f\mathbb R^n\) 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 [2509.01865].

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 \(16\) and \(32\) [2509.01865].

## 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 \((V,\Omega)\) gives rise to a Jordan geometry \(X(V)\) with a canonical \(G_0\)-invariant partial cyclic order whose intervals are affine or projective images of the symmetric cone \(\Omega\). That paper explicitly presents a research program toward “Jordan schemes,” bounded symmetric domains, and order-theoretic geometry [1706.09155].

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 \(p\)-nilpotent operators along \(\pi\)-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 \(\Pi(G)\) and \((r,\mathfrak g)\) [1409.6782].

In birational geometry over finite fields, the phrase is used informally in connection with the Jordan property of groups: a group \(\Gamma\) is Jordan if every finite subgroup \(G\subset \Gamma\) contains a normal abelian subgroup of uniformly bounded index, and \(p\)-Jordan is the characteristic-\(p\) variant
\[
[G:A]\le J\,|G_{(p)}|^e.
\]
That usage concerns subgroup structure rather than coherent Jordan algebras [2605.25788].

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 [1912.04551][2509.04068][2509.01865].

Source: https://www.emergentmind.com/topics/jordan-schemes