ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.10 / 133–150
https://doi.org/10.26493/1855-3974.2405.b43
(Also available at http://amc-journal.eu)
A characterization of exceptional pseudocyclic
association schemes by multidimensional
intersection numbers
Gang Chen * , Jiawei He †
School of Mathematics and Statistics, Central China Normal University, Wuhan, China
Ilia Ponomarenko ‡
Steklov Institute of Mathematics at St. Petersburg, Russia;
Sobolev Institute of Mathematics, Novosibirsk, Russia; and
School of Mathematics and Statistics of Central China Normal University, Wuhan, China
Andrey Vasil’ev §
Sobolev Institute of Mathematics, Novosibirsk, Russia; and
Novosibirsk State University, Novosibirsk, Russia
Received 10 August 2020, accepted 6 April 2021, published online 13 September 2021
Abstract
Recent classification of 32 -transitive permutation groups leaves us with three infinite
families of groups which are neither 2-transitive, nor Frobenius, nor one-dimensional affine.
The groups of the first two families correspond to special actions of PSL(2, q) and PΓL(2, q),
whereas those of the third family are the affine solvable subgroups of AGL(2, q) found by
D. Passman in 1967. The association schemes of the groups in each of these families are
known to be pseudocyclic. It is proved that apart from three particular cases, each of these
exceptional pseudocyclic schemes is characterized up to isomorphism by the tensor of its
3-dimensional intersection numbers.
Keywords: Association schemes, groups, coherent configurations.
Math. Subj. Class. (2020): 05E30, 05E18
*The author was supported by the NSFC grant No. 11971189.
†The author was supported by the NSFC grant No. 11971189.
‡Corresponding author. The author was supported by the Mathematical Center in Akademgorodok under
agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.
§The author was supported by the Mathematical Center in Akademgorodok under agreement No. 075-15-
2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.
E-mail addresses: chengangmath@mail.ccnu.edu.cn (Gang Chen), hjwywh@mails.ccnu.edu.cn (Jiawei He),
inp@pdmi.ras.ru (Ilia Ponomarenko), vasand@math.nsc.ru (Andrey Vasil’ev)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
134 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
1 Introduction
In the late 1960s, H. Wielandt proposed a method for studying permutation groups via
invariant relations. Later, D. Higman axiomatized a part of this method (connected with
binary relations) by introducing a new object called a coherent configuration [6]. The
coherent configuration of a permutation group G is formed by the orbits of the induced
action of G on the Cartesian square of the underlying set of points (for exact definitions,
see Section 2). Looking only at the parameters of this coherent configuration, the so-called
intersection numbers, one can easily determine whether the original group is transitive,
primitive, 2-transitive, etc. For example, the transitivity of a group G means exactly that
the coherent configuration of G is an association scheme.
The concept of pseudocyclic (association) scheme goes back to research of D. Mes-
ner [11], related with constructing of designs and strongly regular graphs; the defining
property of such a scheme is that the ratio of the multiplicity and degree of its nonprin-
cipal irreducible character does not depend on the choice of the character. It was proved
in [13, Theorem 3.2] that this is equivalent to a certain relation for intersection numbers,
see Subsection 2.6.
The class of pseudocyclic schemes contains all Frobenius schemes, i.e., the coherent
configurations of the Frobenius groups, and, moreover, every pseudocyclic scheme of rank
sufficiently large comparing with its degree is Frobenius [13, Theorem 1.1]. Thus the
pseudocyclic schemes can be considered as combinatorial analogs of the Frobenius groups.
It should be mentioned that the analogy is not complete, because there exist schurian (i.e.,
those associated with permutation groups) pseudocyclic schemes which are not Frobenius,
as well as non-schurian pseudocyclic schemes [2, Example 2.6.15].
It is well known that every Frobenius group is 32 -transitive, i.e., is transitive and all the
orbits of the stabilizer of a point α, other than {α}, have the same size greater than 1. It
immediately follows that so is the automorphism group of any Frobenius scheme. More-
over, the above mentioned relation for intersection numbers implies that the automorphism
group of a schurian pseudocyclic scheme is also 32 -transitive. A recent classification of
3
2 -transitive permutation groups shows that in most cases the coherent configuration of a
3
2 -transitive group is pseudocyclic, see Subsection 6.1. We cite a part of this classification
in the following theorem, see [10, Corollaries 2,3].
Theorem 1.1. Let G be a 32 -transitive permutation group of degree n. Assume that neither
G is 2-transitive or Frobenius nor G ≤ AΓL(1, q) for some prime power q. Then apart
from finitely many cases,
(1) n = q(q − 1)/2, q = 2d ≥ 8, and either G = PSL(2, q), or d is prime and
G = PΓL(2, q),
(2) n = q2, q is odd, and G ≤ AGL(2, q) is the affine group with point stabilizer of
order 4(q − 1), consisting of all monomial matrices of determinant ±1.
Remark 1.2. The finitely many cases mentioned in Theorem 1.1 include some affine
groups of degree at most 134 and the groups Alt(7) and Sym(7) both of degree 21.
The association schemes of the groups in statement (1) of Theorem 1.1 appeared in
Master Thesis of H. Hollmann (1982); such a scheme is called a large or small Hollmann
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 135
scheme depending on whether G = PSL(2, q) or G = PΓL(2, q).1 These schemes have
been studied in [8]; in particular, it was proved there that both are pseudocyclic. The group
in statement (2) of Theorem 1.1 appeared in D. Passman’s characterization of solvable 32 -
transitive groups [16]. The association scheme of this group, the Passman scheme, is also
pseudocyclic [13].
The goal of the present paper is to establish combinatorial characterizations of the Holl-
mann and Passman schemes; from the point of view of Theorem 1.1, they can naturally be
considered as exceptional. One of the best possible combinatorial characterization of an
association scheme is obtained when the scheme in question is determined up to (combina-
torial) isomorphism by its intersection numbers; in this case the scheme is called separable.
However, most of association schemes are not separable. In [4], multidimensional inter-
section numbers and separability number s(X ) of a coherent configuration X have been
introduced and studied (see also [2, Section 3.5 and 4.2]). According to the definition,
s(X ) ≤ m if and only if X is determined up to isomorphism by its m-dimensional inter-
section numbers; thus s(X ) = 1 if and only if X is separable. It was proved in [4] that
s(X ) = 1 or 2 if X is the scheme of a Hamming, Johnson, or Grassmann graph; later, the
estimate s(X ) ≤ 3 has been established in [5] for any cyclotomic scheme X over finite
field.
Theorem 1.3. Let X be a large Hollmann scheme. Then s(X ) ≤ 2.
The proof of Theorem 1.3 is given in Section 3. The difficult step in the proof is to
verify that the one point extension of the scheme X (which is a combinatorial analog of a
one point stabilizer of a permutation group) is a coherent configuration of the stabilizer of
this point in Aut(X ). In proving this fact we use the formulas for the intersection numbers
of X , which were calculated in [8].
The proofs of the following two theorems are based on Theorem 4.1 (see Section 4),
giving a sufficient condition for an arbitrary coherent configuration X to be partly regular,
i.e., to be the coherent configuration of a permutation group having a faithful regular orbit.
Using this sufficient condition we are able to show that if X is the small Hollmann scheme
(apart from several exceptions) or the Passman scheme, then a two point extension of X is
partly regular. Modulo known results, this immediately implies that s(X ) ≤ 3.
Theorem 1.4. Let X be a small Hollmann scheme of degree q(q − 1)/2, where q = 2d
with prime d ̸= 7, 11, 13. Then s(X ) ≤ 3.
In the three exceptional cases of Theorem 1.4, the sufficient condition given in Theo-
rem 4.1 does not work. It seems that the conclusion of Theorem 1.4 is also true for them.
However, the corresponding schemes are too large to check this statement by a direct cal-
culation.
Theorem 1.5. Let X be a Passman scheme. Then s(X ) ≤ 3.
Throughout the paper, we actively use the notation, concepts, and statements from the
theory of coherent configurations. All of them can be found in the monograph [2]. In
Section 2, we give a brief extract from the theory of coherent configurations that is relevant
for this paper.
1According to the Galois correspondence between permutation groups and coherent configurations [2, Sec-
tion 2.2], the smaller groups correspond to larger coherent configurations
136 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
The authors are grateful to the anonymous referees whose valuable comments helped
to improve the presentation and eliminate inaccuracies, and to Saveliy Skresanov for his
supplement to the GAP-package COCO2P, that was used for the computations related to
Theorem 5.5.
Notation.
• For a prime power q, Fq is a finite field of order q.
• Throughout the paper, Ω is a finite set.
• The diagonal of the Cartesian product Ω × Ω is denoted by 1Ω; for α ∈ Ω, we
set 1α := 1{α}.
• For r ⊆ Ω×Ω, we set r∗ = {(β, α) : (α, β) ∈ r} and αr = {β ∈ Ω : (α, β) ∈ r},
α ∈ Ω.
• For relations r, s ⊆ Ω× Ω, we set r · s = {(α, β) : (α, γ) ∈ r, (γ, β) ∈ s}.
• For a set S of relations on Ω, we define S∗ = {s∗ : s ∈ S} and put S∪ to be the set
of all unions of the relations of S.
2 Coherent configurations
2.1 Rainbows
Let Ω be a finite set and S a partition of Ω × Ω. A pair X = (Ω, S) is called a rainbow
on Ω if
1Ω ∈ S∪, and S∗ = S.
The elements of the sets Ω, S = S(X ), and S∪ are called, respectively, the points, basis
relations, and relations of X . The numbers |Ω| and |S| are called the degree and rank of X ,
respectively. The unique basis relation containing a pair (α, β) ∈ Ω × Ω is denoted by
rX (α, β); we omit the subscript X wherever it does not lead to misunderstanding.
A set ∆ ⊆ Ω is called a fiber of the rainbow X if 1∆ ∈ S; the set of all fibers is denoted
by F := F (X ). The point set Ω is the disjoint union of fibers. If ∆ is a union of fibers,
then the pair
X∆ = (∆, S∆)
is a rainbow, where S∆ consists of all s∆ = s ∩ (∆×∆), s ∈ S.
Let X = (Ω, S) and X ′ = (Ω′, S′) be rainbows. A bijection f : Ω → Ω′ is called
a combinatorial isomorphism (or simply isomorphism) from X to X ′ if Sf = S′, where
Sf = {sf : s ∈ S}. When X = X ′, the set of all these isomorphisms form a permutation
group on Ω. This group has a (normal) subgroup
Aut(X ) = {f ∈ Sym(Ω): sf = s for all s ∈ S},
called the automorphism group of X .
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 137
2.2 Coherent configurations
A rainbow X = (Ω, S) is called a coherent configuration if for any r, s, t ∈ S, the number
ctrs = |αr ∩ βs∗|
does not depend on the choice of (α, β) ∈ t; the numbers ctrs are called the intersection
numbers of X . If, in addition, 1Ω ∈ S, then the coherent configuration X is said to be
homogeneous, an association scheme, or just a scheme. A scheme X is called symmetric if
s = s∗ for all s ∈ S.
Let X be a coherent configuration. Then for any s ∈ S, there exist uniquely determined
∆,Γ ∈ F such that s ⊆ ∆× Γ. Denote by S∆,Γ the set of all s contained in ∆× Γ. Then
the union
S =
⋃
∆,Γ∈F
S∆,Γ
is disjoint. The positive integer |δs|, δ ∈ ∆, equals the intersection number c1∆ss∗ , and hence
does not depend on the choice of δ. It is called the valency of s and denoted by ns. In
homogeneous case, ns = ns∗ and also
ntc
t∗
rs = nrc
r∗
st = nsc
s∗
tr , r, s, t ∈ S. (2.1)
A basis relation s ∈ S is called a matching if ns = ns∗ = 1; note that s is not
necessarily symmetric like in theory of undirected graphs. The matching s ∈ S∆,Γ defines
a bijection from ∆ to Γ, taking δ ∈ ∆ to the unique point of the singleton δs. Furthermore,
one can see that if r ∈ S and t = s · r (respectively, t = r · s) is nonempty, then t ∈ S.
Let G be a permutation group on Ω. Denote by (α, β)G the orbit of the induced action
of G on Ω× Ω, that contains the pair (α, β). Then
Inv(G) = Inv(G,Ω) = (Ω, {(α, β)G : α, β ∈ Ω})
is a coherent configuration; we say that Inv(G) is the coherent configuration associated
with G. A coherent configuration X is said to be schurian if X = Inv(Aut(X )).
2.3 Separability
Let X = (Ω, S) and X ′ = (Ω′, S′) be coherent configurations. A bijection φ : S →
S′, s 7→ s′, is called an algebraic isomorphism from X to X ′ if
ctrs = c
t′
r′s′ , r, s, t ∈ S. (2.2)
When X = X ′, the set of all such φ forms a subgroup of Sym(S), denoted by Autalg(X ).
Each isomorphism f from X to X ′ induces an algebraic isomorphism from X to X ′,
which maps r ∈ S to rf ∈ S′. A coherent configuration X is said to be separable if every
algebraic isomorphism from X is induced by a suitable bijection (which is an isomorphism
of the coherent configurations in question).
The algebraic isomorphism φ induces a bijection from S∪ to (S′)∪: the union r∪s∪· · ·
of basis relations of X is mapped to r′ ∪ s′ ∪ · · · . This bijection is also denoted by φ. It
preserves the dot product, i.e., φ(r · s) = φ(r) · φ(s) for all r, s ∈ S.
138 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
2.4 Coherent closure
There is a natural partial order ≤ on the set of all rainbows on the same set Ω. Namely,
given two such rainbows X and X ′, we set
X ≤ X ′ ⇔ S(X )∪ ⊆ S(X ′)∪.
The minimal and maximal elements with respect to this order are the trivial and discrete
coherent configurations, respectively: the basis relations of the former are the reflexive
relation 1Ω and (if |Ω| > 1) its complement in Ω × Ω, whereas the basis relations of the
latter are singletons.
The functors X → Aut(X ) and G → Inv(G) form a Galois correspondence between
the posets of coherent configurations and permutation groups on the same set, i.e.,
Y ≤ X ⇒ Aut(Y) ≥ Aut(X ) and L ≤ K ⇒ Inv(L) ≥ Inv(K),
and
Aut(Inv(Aut(X ))) = Aut(X ) and Inv(Aut(Inv(G))) = Inv(G).
The coherent closure WL(T ) of a set T of relations on Ω, is defined to be the smallest
coherent configuration on Ω, for which T is a set of relations. The point extension Xα,β,...
of the rainbow X with respect to the points α, β, . . . ∈ Ω is defined to be WL(T ), where T
consists of S(X ) and the relations 1α, 1β , . . .. In other words, Xα,β,... is the smallest co-
herent configuration on Ω that is larger than or equal to X and has singletons {α}, {β}, . . .
as fibers.
2.5 Multidimensional intersection numbers
The theory of multidimensional extensions of coherent configurations has been developed
in [4], see also [2, Section 3.5].
Let m ≥ 1 be an integer. The m-extension of a coherent configuration X on Ω is
defined to be the smallest coherent configuration on Ωm, which contains the Cartesian m-
power of X and for which the set Diag(Ωm) is a union of fibers. The intersection numbers
of the m-extension are called the m-dimensional intersection numbers of the configura-
tion X . If m = 1, then the m-extension of X coincides with X and the m-dimensional
intersection numbers of X are the ordinary intersection numbers.
An algebraic isomorphism φ from X to X ′ is said to be m-dimensional if it can be
extended to an algebraic isomorphism from the m-extension of X to that of X ′, that takes
Diag(Ωm) to Diag(Ω′m). The separability number s(X ) of the coherent configuration X
is defined to be the smallest positive integer m for which every m-dimensional algebraic
isomorphism from X is induced by some isomorphism. Thus, the equality s(X ) = m ex-
presses the fact that X is determined up to isomorphism by its tensor of the m-dimensional
intersection numbers. The following statement was proved in [4, Theorem 4.6(1)].
Lemma 2.1. Let X be a coherent configuration. Then s(X ) ≤ s(Xα) + 1 for any point α
of X .
2.6 Pseudocyclic schemes
Let X = (Ω, S) be a coherent configuration. The indistinguishing number of a relation
s ∈ S(X ) is defined to be the sum c(s) of the intersection numbers csrr∗ , r ∈ S. For each
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 139
pair (α, β) ∈ s, we have c(s) = |c(α, β)|, where
c(α, β) = {γ ∈ Ω : r(γ, α) = r(γ, β)}. (2.3)
The maximum c(X ) of the numbers c(s), where s runs over the set of all irreflexive basis
relations of X , is called the indistinguishing number of X . It is easily seen that c = 0 if
and only if ns = 1 for each s ∈ S.
Assume that X is a scheme. In accordance with [13, Theorem 3.2], X is pseudocyclic
of valency k if the equalities
c(s) + 1 = k = ns
hold for all irreflexive s ∈ S. The class of pseudocyclic schemes includes all (homoge-
neous) coherent configurations associated with regular or Frobenius groups.
2.7 Partly regular coherent configurations
A coherent configuration X is said to be partly regular if there exists a point α ∈ Ω such
that |αs| ≤ 1 for all s ∈ S; the point α is said to be regular. When all the points of X are
regular, we say that X is semiregular, and regular if X is a scheme. Thus, X is semiregular
if and only if c(X ) = 0.
The following statement taken from [2, Theorem 3.3.19] shows, in particular, that the
partly regular (respectively, semiregular, regular) coherent configurations are in one-to-one
correspondence with those of the form Inv(G), where G is a permutation group having a
faithful orbit (respectively, G is semiregular, regular).
Theorem 2.2. Every partly regular coherent configuration X is schurian and separable.
In particular, s(X ) = 1.
The key point in the proof of Theorem 2.2 is the lemma below [2, Lemma 3.3.20]; it is
also used in the proof of Theorem 1.3.
Lemma 2.3. Let X be a coherent configuration and ∆ a union of fibers of X . Assume that
for every Γ ∈ F there exists s ∈ S∆,Γ such that ns = 1. Then
(1) the restriction mapping Aut(X ) → Aut(X∆) is a group isomorphism,
(2) X is schurian and separable whenever X∆ is schurian and separable.
3 Large Hollmann schemes
3.1 General properties
Throughout this section, d ≥ 3 is an integer, q = 2d, and G = PSL(2, q) the permutation
group of degree n = q(q − 1)/2 from Theorem 1.1(1). The lemma below immediately
follows from Theorem 1.2(iii) and Lemma 6.2 proved in [1].
Lemma 3.1. For any point α, we have Gα = D2(q+1). Moreover,
|∆| = q + 1, ∆ ∈ Orb(Gα), ∆ ̸= {α}. (3.1)
To study the large Hollmann scheme X = Inv(G), we make use of some results proved
in [8]. However, formally, the definition of the scheme given there is different from the
140 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
definition of X . Thus our first goal is to verify that X is exactly the symmetric pseudocyclic
scheme X ′ of degree n and valency q + 1 defined in [13] and studied in [8].
First, we note that Aut(X ) is a 32 -transitive group of the same degree as G; in particular,
Aut(X ) is not a subgroup of AΓL(1, r) for some r. Moreover, it is neither 2-transitive,
nor Frobenius (because G ≤ Aut(X )). By Theorem 1.1 and Remark 1.2, this implies that
Aut(X ) = PSL(2, q) or PΓL(2, q). The latter case is impossible by Lemma 3.1, because
G and Aut(X ) have the same subdegrees. Consequently, G = Aut(X ). On the other
hand, the scheme X ′ is also associated with G. Therefore a similar argument shows that
Aut(X ′) = G = Aut(X ). Thus,
X ′ = Inv(Aut(X ′)) = Inv(Aut(X )) = X . (3.2)
Proposition 3.2. The large Hollmann scheme X is symmetric and pseudocyclic of degree
q(q − 1)/2, rank q/2, and valency q + 1. Moreover,
Aut(X ) = G and Aut(Xα) = Gα = D2(q+1) for all α. (3.3)
Proof. By the remark before the proposition, we need to verify the second equality in (3.3)
only. By [2, Proposition 3.3.3(1)], we have Aut(Xα) = Aut(X )α. Thus the required
statement immediately follows from the first equality in (3.3) and Lemma 3.1.
Equality (3.2) allows us to use formulas for the intersection numbers of the scheme X ′,
given in [8, Theorem 2.2]. Namely, let S = S(X ) and
T0 = {x ∈ F2d : Tr(x) = 0},
where Tr(x) is the trace of x over the prime subfield of the field F2d . Then there is a
bijection T0 → S, x 7→ sx, such that s0 is reflexive and
cszsx,sy = 1 ⇔ Tr(xz) = 0 and x+ y + z = 0. (3.4)
As is easily seen, T0 is a linear space of dimension d− 1 over F2.
3.2 One point extension
Let us analyze the extension Xα of the large Hollmann scheme X with respect to a point α.
Since the scheme X is schurian, each fiber of the coherent configuration Xα is of the form
∆ = αs for some s ∈ S [2, Theorem 3.3.7]. When s = sx for some x ∈ T0, the fiber ∆
is denoted by ∆x. Thus,
F (Xα) = {∆x : x ∈ T0}.
Theorem 3.3. Let x and y be nonzero elements of T0. Then the set S(Xα)∆x,∆y contains
a matching.
Proof. For d = 3, the statement has been checked with the help of the computer pack-
age COCO2P [9]. Assume that d ≥ 4. We need auxiliary lemmas.
Lemma 3.4. Theorem 3.3 holds whenever Tr(xy) = 0.
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 141
Proof. Let z = x+ y. Then obviously z ∈ T0. Moreover
Tr(xz) = Tr(x2 + xy) = Tr(x2) + Tr(xy) = 0.
By formula (3.4), this implies that cszsx,sy = 1. If z = 0, then x = y and 1∆x is a desired
matching. Assume that z is nonzero. Then nsx = nsz , because X is a pseudocyclic scheme
(Proposition 3.2). Since X is also symmetric, we have
csxsy,sz =
nsz
nsx
cszsx,sy = 1,
see (2.1). Therefore if r = sz ∩ (αsx × αsy), then |βr| = 1 for all β ∈ ∆x, and |βr∗| = 1
for all β ∈ ∆y (here we use the fact that |∆x| = |∆y|). Since r is a relation of Xα [2,
Lemma 3.3.5], this implies that r belongs to S(Xα)∆x,∆y . Thus, r is a required matching.
Let us define a graph X with nonzero elements of T0 as the vertices and in which two
distinct vertices x and y are adjacent if and only if Tr(xy) = 0. One can see that X is an
undirected graph with exactly |T0| − 1 vertices.
Lemma 3.5. The graph X is connected.
Proof. Given x ∈ T0, we define the linear mapping
fx : T0 → F2, y 7→ Tr(xy).
Now let x ̸= y be two vertices of the graph X. If Tr(xy) = 0, then x and y are connected by
an edge. Let Tr(xy) ̸= 0. Then both fx and fy are nonzero linear mappings, and ker(fx)
and ker(fy) are subspaces of T0 of codimension one. Therefore, ker(fx) ∩ ker(fy) is a
subspace of T0 of codimension at most two. Consequently,
dim(ker(fx) ∩ ker(fy)) ≥ dim(T0)− 2 = d− 3 ≥ 1.
It follows that ker(fx)∩ker(fy) contains at least one nonzero vector, say z. Then Tr(xz) =
Tr(yz) = 0. Since Tr(xy) ̸= 0, this implies that z ̸= x, y. Thus the vertices x, y are at
distance two in the graph X, in particular, X is connected.
Let us return to the proof of Theorem 3.3. By Lemma 3.5, the vertices x and y of the
graph X are connected by a path
x = x0, x1, . . . , xk = y,
where k ≥ 1. For i = 0, . . . , k − 1, the vertices xi and xi+1 are adjacent and hence
Tr(xixi+1) = 0. Denote by si the matching in S(Xα)∆xi ,∆xi+1 the existence of which is
guaranteed by Lemma 3.4. Then the dot product
s = s0 · s1 · · · sk−1
is a desired matching belonging to S(Xα)∆x,∆y .
Corollary 3.6. Let ∆ = ∆x for nonzero x ∈ T0. Then the coherent configuration Y =
(Xα)∆ is schurian and separable. Moreover, the extension of Y with respect to at least one
point is partly regular.
142 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
Proof. By Proposition 3.2, we have Aut(Xα) = D2(q+1). Furthermore, the hypothesis of
Lemma 2.3 is satisfied for X = Xα by Theorem 3.3. Thus by statement (1) of that lemma,
we have
H := Aut(Y) ∼= Aut(Xα) = D2(q+1). (3.5)
On the other hand, |∆| = q + 1 by formula (3.1). Consequently, the group H contains a
normal regular cyclic subgroup C of order q + 1. In terms of [2, Section 4.4], this means
that Y is isomorphic to a normal circulant scheme. The radical of such a scheme, being a
subgroup of the group C, is of order at most 2; this follows from the implication (1)⇔(3)
in [5, Theorem 6.1]. Since the number |C| = q + 1 = 2d + 1 is odd, the radical is trivial.
Thus, the scheme Y is schurian by [2, Corollary 4.4.3], and every its extension with respect
to at least one point is partly regular by [2, Theorem 4.4.7].
It remains to verify that Y is separable. Since Y is schurian by above, we have Y =
Inv(Aut(Y)) = Inv(H). By virtue of (3.5), this means that Y is the coherent configuration
associated with D2(q+1). Thus, the required statement follows from [2, Exercise 2.7.33].
3.3 Proof of Theorem 1.3
By Lemma 2.1, it suffices to verify that a one point extension of a large Hollmann scheme
is separable. But this immediately follows from Theorem 3.7 below.
Theorem 3.7. Let q = 2d where d ≥ 3. Then the extension of the large Hollmann
scheme of degree q(q − 1)/2 with respect to at least one point is schurian and separa-
ble.
Proof. Let X ′ be the extension of the large Hollmann scheme X with respect to m ≥ 1
points α = α1, α2, . . . , αm. Let x ∈ T0 be nonzero and ∆ = ∆x. Then the hypothesis of
Lemma 2.3 is satisfied for X = X ′. Indeed, each Γ ∈ F (X ′) other than {α} is contained
in ∆y for some nonzero y ∈ T0. By Theorem 3.3, there is a matching s′ ∈ S(Xα)∆x,∆y ,
and as the required relation s one can take s′ ∩ (∆× Γ).
By Lemma 2.3(2), the coherent configuration X ′ is schurian and separable whenever
so is X ′∆. If m = 1, then X ′ = Xα and we are done by Corollary 3.6. Let m > 1. We
claim that there exist β2, . . . , βm ∈ ∆ such that
X ′ = Xα,β2,...,βm . (3.6)
Indeed, without loss of generality we may assume that none of the αi, i > 1, equals α. By
Theorem 3.3, there is a matching si ∈ S(Xα)∆i,∆, where ∆i is the fiber of Xα, contain-
ing αi. Then αisi = {βi} for some βi ∈ ∆. It follows that for any extension of Xα, each
or none of the two singletons {αi} and {βi} is a fiber of this extension, see [2, Corollary
3.3.6]. Thus,
X ′ = Xα,α2,...,αm = Xα,β2,...,βm ,
which completes the proof of the claim.
Now from formula (3.6) and the fact that βi ∈ ∆ for all i, it easily follows that
X ′∆ =(Xα,α2,...,αm)∆ = (Xα,β2,...,βm)∆
=((Xα,β2,...,βm)∆)β2,...,βm ≥ ((Xα)∆)β2,...,βm .
The coherent configuration on the right-hand side of this relation is partly regular by Corol-
lary 3.6. Therefore, the coherent configuration X ′∆ is also partly regular. By Theorem 2.2,
this implies that X ′∆ is schurian and separable, as required.
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 143
4 A lower bound for indistinguishing number
The main result of this section (Theorem 4.1 below) establishes a lower bound for the
indistinguishing number of a coherent configuration which is not partly regular, cf. [17,
Theorem 3.1]. This bound gives a sufficient condition for a coherent configuration to be
partly regular, and is used to prove Theorems 1.4 and 1.5 in the next section.
Theorem 4.1. Let X be a coherent configuration of degree n, k the maximal cardinality of
a fiber of X , and c = c(X ). If X is not partly regular, then
(2k − 1)c ≥ n. (4.1)
Proof. Let X = (Ω, S). In the sequel, ∆ ∈ F (X ) and |∆| = k. The fiber ∆ contains
at least two points, for otherwise k = 1 and X is the discrete and hence partly regular
configuration in contrast to the hypothesis of the theorem. Set
∆α = {δ ∈ ∆ : nr(α,δ) = 1} and Ω1 = {α ∈ Ω : ∆α = ∆}.
Lemma 4.2. Ω1 is a (possibly empty) union of fibers of cardinality k. Moreover, the co-
herent configuration XΩ1 is semiregular.
Proof. Let Γ be a fiber containing a point of Ω1. The set SΓ,∆ consists of s = r(γ, δ),
where γ ∈ Γ ∩ Ω1 and δ runs over ∆. Consequently, ns = 1 for all s ∈ SΓ,∆. Therefore,
Γ ⊆ Ω1. Thus, Ω1 is a union of fibers of X . Furthermore, given s ∈ SΓ,∆ we have
k ≥ |Γ| = ns |Γ| = ns∗ |∆| ≥ |∆| = k.
This proves the first statement. To prove the second one, let ∆′ and ∆′′ be fibers contained
in Ω1. By the definition of Ω1 and the first statement, any relations s′ ∈ S∆′,∆ and s′′ ∈
S∆,∆′′ are matchings. Therefore, s′ · s′′ is a matching contained in S∆′,∆′′ . Thus, SΩ1
consists of matchings and we are done.
By Lemma 4.2 and the hypothesis of the theorem, the complement Ω′ of the set Ω1 in Ω
contains at least two distinct points.
Lemma 4.3. For each γ ∈ Ω′, ∑
s∈Sγ
ns ≥
k
2
,
where Sγ = {r(γ, δ) : δ ∈ ∆ and nr(γ,δ) > 1}.
Proof. We have ∑
s∈Sγ
ns =
∑
s∈Sγ
|γs| = |∆| − |∆γ |. (4.2)
Since |∆| = k, this proves the required inequality if ∆γ = ∅. Let δ ∈ ∆γ . It is easily seen
that r(δ, λ) = r(δ, γ) · r(γ, λ) is a matching of X∆ for each λ ∈ ∆γ . Therefore,
∆γ ⊆ {λ ∈ ∆ : nr(δ,λ) = 1}. (4.3)
On the other hand, denote by e the union of all matchings of the scheme X∆. Then e
is a relation of this scheme. Moreover, e is an equivalence relation on ∆ (see [2, Theo-
rem 2.1.25(4)]) and the set on the right-hand side of (4.3) is a class of e. In view of [2,
144 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
Corollary 2.1.23], the cardinality a of this class divides |∆| = k. Furthermore, a ̸= k, for
otherwise, ∆γ = ∆ and then γ ∈ Ω1, a contradiction. Thus,
|∆γ | ≤ a ≤
k
2
and the required statement follows from (4.2).
Lemma 4.4. Let ε = 2(k−1)2k−1 . Assume that |Ω
′| ≥ εn. Then inequality (4.1) holds.
Proof. Denote by N the cardinality of the set
{(α, β, γ) ∈ ∆×∆× Ω′ : α ̸= β, γ ∈ c(α, β)}, (4.4)
where c(α, β) is as in formula (2.3). The number of (α, β) ∈ ∆×∆ with α ̸= β is equal
to k(k − 1). Therefore there exists at least one such pair for which
|c(α, β)| ≥ N
k(k − 1)
. (4.5)
On the other hand, let γ ∈ Ω′, and let Sγ be as in Lemma 4.3. Then ns ≥ 2 for
all s ∈ Sγ . For every such s there are exactly ns(ns − 1) triples (α, β, γ) with distinct
α, β ∈ γs, and all these triples belong to the set (4.4). By Lemma 4.3 this implies that
N =
∑
γ∈Ω′
∑
s∈SΓ,∆
ns(ns − 1) ≥
∑
γ∈Ω′
∑
s∈Sγ
ns ≥
∑
γ∈Ω′
k
2
=
|Ω′| k
2
,
where Γ is the fiber containing γ. By formula (4.5) and the lemma assumption, we obtain
c ≥ |c(α, β)| ≥ N
k(k − 1)
≥ |Ω
′|
2(k − 1)
≥ 2(k − 1)n
2k − 1
· 1
2(k − 1)
=
n
2k − 1
,
as required.
By Lemma 4.4, we may assume that |Ω′| < εn. The coherent configuration X is not
partly regular. Therefore no point δ ∈ Ω1 is regular and there exist distinct α, β ∈ Ω′ such
that δ ∈ c(α, β). Since the coherent configuration XΩ1 is semiregular (Lemma 4.2), the
relation s = r(δ, λ) is a matching for all λ ∈ Ω1. It follows that
r(α, λ) = r(α, δ) · s = r(β, δ) · s = r(β, λ).
Consequently, Ω1 ⊆ c(α, β). This implies that
c ≥ |c(α, β)| ≥ |Ω1| = n− εn > n
(
1− 2(k − 1)
2k − 1
)
=
n
2k − 1
which completes the proof of Theorem 4.1.
Corollary 4.5. Let X be a coherent configuration of degree n, c = c(X ), and t an irreflex-
ive basis relation of X . Assume that (2mt − 1) c < n, where
mt = max
r,s∈S
ctrs. (4.6)
Then the extension of X with respect to any two points forming a pair from t is partly
regular.
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 145
Proof. Let X ′ be the extension of X with respect to the points α, β such that (α, β) ∈ t.
Then each fiber ∆ of X ′ different from both {α} and {β} is contained in the set αr ∩ βs∗
for appropriate r, s ∈ S. It follows that
|∆| ≤ |αr ∩ βs∗| = ctrs ≤ mt.
Thus the maximal cardinality k′ of a fiber of X ′ is less than or equal to mt. Since obviously
c′ = c(X ′) is less than or equal to c, the condition of the corollary implies that
(2k′ − 1) c′ ≤ (2mt − 1) c < n.
Thus X ′ is partly regular by Theorem 4.1.
5 Small Hollmann and Passman schemes
5.1 Algebraic fusion
Let X = (Ω, S) be a coherent configuration, and let Φ be a group of algebraic automor-
phisms of X . For each s ∈ S, set
sΦ =
⋃
φ∈Φ
φ(s).
Clearly, (1Ω)Φ = 1Ω. Moreover the set SΦ = {sΦ : s ∈ S} forms a partition of the
Cartesian square Ω2. According to [2, Lemma 2.3.26], the pair
XΦ = (Ω, SΦ)
is a coherent configuration called the algebraic fusion of X with respect to Φ. In the
following lemma, we establish a simple upper bound for the intersection numbers of an
algebraic fusion.
Lemma 5.1. In the above notation, let r, s, t ∈ S and mt be as in (4.6). Then
ct
Φ
rΦsΦ ≤ mt |Φ|
2.
Proof. We have ct
Φ
rΦsΦ =
∑
φ,ψ∈Φ
ctφ(r)ψ(s) ≤ mt |Φ|
2.
5.2 Proof of Theorem 1.4
Let X be a small Hollmann scheme. Then the degree of X is equal to n = q(q − 1)/2,
where q = 2d for a prime d ≥ 3. Moreover, X is associated with the permutation group
G = PΓL(2, q) of degree n from Theorem 1.1(1). As in Subsection 3.1, one can see that
X coincides with symmetric pseudocyclic scheme X ′ of degree n and valency d(q + 1),
associated with the group PΓL(2, q) and studied in [8]. In particular, X is obtained from
the large Hollmann scheme of degree n by merging the basis relations via the Frobenius
map x 7→ x2, x ∈ Fq . In other words, X is the algebraic fusion of the large Hollmann
scheme of degree n with respect the induced action of Aut(Fq) on its basis relations.
Proposition 5.2. Let X and Xq be the small and large Hollmann schemes of degree q(q −
1)/2, respectively. Then
146 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
(1) X = XqΦ, where Φ ≤ Autalg(Xq) is a group of order d,
(2) X is a pseudocyclic scheme of valency d(q + 1),
(3) for each irreflexive t ∈ S(X ), we have mt ≤ 4d2, where mt is as in (4.6).
Proof. Statements (1) and (2) follow from the above discussion. Next, from the formulas
for the intersection numbers of the scheme Xq , given in [8, Theorem 2.2], it follows that
czxy ≤ 4 for all irreflexive x, y, z ∈ S(Xq). In particular,
mz ≤ 4.
On the other hand, by statement (1), each irreflexive t ∈ S(X ) is of the form zΦ for some
irreflexive z. Thus by Lemma 5.1,
mt = max
x,y∈S(Xq)
cz
Φ
xΦyΦ ≤ mz|Φ|
2 ≤ 4d2,
which proves statement (3).
Let us prove Theorem 1.4. If d = 3, then a straightforward calculation shows that X is
trivial scheme and hence c(X ) = 1. Let d > 3. By Lemma 2.1 and Theorem 2.2, it suffices
to verify that the extension of X with respect to at least two points is partly regular.
Theorem 5.3. Let q = 2d, where d > 3 is a prime and d ̸= 7, 11, 13. Then every extension
of the small Hollmann scheme of degree q(q − 1)/2 with respect to at least two points is
partly regular.
Proof. Let X be the small Hollmann scheme of degree n = q(q − 1)/2. By Proposi-
tion 5.2(2), the number c = c(X ) is equal to d(2d + 1) − 1. By statement (3) of the same
proposition, mt ≤ 4 d2 for any irreflexive t ∈ S(X ). Now if d > 16, then
(2mt − 1) c < (8d2 − 1) (d (2d + 1)− 1) < 2d−1(2d − 1) = n.
By Corollary 4.5, this proves the required statement for all (prime) d > 13. In the remain-
ing case, d = 5, the required statement has been checked with the help of the computer
package COCO2P [9].
5.3 Proof of Theorem 1.5
Let q be an odd prime power. The permutation group G ≤ AGL(2, q) defined in Theo-
rem 1.1(2) has a Frobenius subgroup H consisting of the permutations(
x
y
)
→
(
a 0
0 a−1
)(
x
y
)
+
(
b
c
)
, x, y ∈ Fq, (5.1)
where a, b, c ∈ Fq and a ̸= 0. The group D ≤ GL(2, q) consisting of the eight matrices(
±1 0
0 ±1
)
and
(
0 ±1
±1 0
)
is contained in G, normalizes H , and, moreover, G = H ⋊ D. In particular, D acts
on the basis relations of the Frobenius scheme Y = Inv(H) as a group Φ of algebraic
automorphisms.
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 147
Proposition 5.4. Let X = Inv(G) be the Passman scheme of degree q2. Then
(1) X = YΦ, where Φ ≤ Autalg(Y) is a group of order 2,
(2) X is a pseudocyclic scheme of valency 2(q − 1),
(3) there is an irreflexive t ∈ S(X ) such that mt ≤ 8, where mt is as in (4.6).
Proof. Statements (1) and (2) follow from [13, Section 4.3]. To prove statement (3), let u
be the basis relation of Y , containing the pair (α, β), where
α = (0, 0) and β = (1, 1).
It suffices to verify that mu ≤ 2; indeed, then t = uΦ is the required relation by state-
ment (1) and Lemma 5.1.
We need to verify that curs ≤ 2 for all r, s ∈ S(Y). Without loss of generality, we may
assume that r and s are such that curs ̸= 0. Then there is γ ∈ αr ∩ βs∗. It follows that
αr = γHα , βs∗ = γHβ , and
curs = |γHα ∩ γHβ |. (5.2)
Let us calculate the number on the right-hand side. Using the explicit form (5.1) of the
elements of H , one can easily find that the groups Hα and Hβ consist of permutations the
parameters a, b, c of which satisfy the relations
b = c = 0 and a+ b = a−1 + c = 1,
respectively. Consequently, assuming γ = (x, y), we have
γHα =
{(
ax,
y
a
)
: a ∈ F∗q
}
and γHβ =
{(
a′x+ 1− a′, y
a′
+ 1− 1
a′
)
: a′ ∈ F∗q
}
.
In view of (5.2), the intersection number curs is equal to the number of elements a ∈ F∗q
such that
ax = a′x+ 1− a′ and y
a
=
y
a′
+ 1− 1
a′
.
If exactly one of x, y equals 0, then these equations are satisfied for a = 1 = a′ only, and
curs = 1. Assume that x ̸= 0 ̸= y. Then
a = a′
(
1− 1
x
)
+
1
x
and
1
a
=
1
a′
(
1− 1
y
)
+
1
y
,
and hence
1 =
(
1− 1
x
)(
1− 1
y
)
+
a′
y
(
1− 1
x
)
+
1
a′x
(
1− 1
y
)
+
1
xy
.
This gives a quadratic equation with unknown a′, and at most two possible values for a′.
Thus, curs ≤ 2.
Let us prove Theorem 1.5. By Lemma 2.1 and Theorem 2.2, it suffices to verify that
the extension of X with respect to at least two points is partly regular.
148 Ars Math. Contemp. 21 (2021) #P1.10 / 133–150
Theorem 5.5. Let q be an odd prime power. Then every extension of the Passman scheme
of degree q2 with respect to at least two points is partly regular.
Proof. Let X be the Passman scheme of degree n = q2. By Proposition 5.4(2), the number
c = c(X ) is equal to 2(q − 1) − 1. Let t be the basis relation of X , defined in Proposi-
tion 5.4(3). Then mt ≤ 8. Now if q ≥ 29, then
(2mt − 1) c ≤ 15 (2q − 3) < q2 = n.
By Corollary 4.5, this proves the required statement for all q ≥ 29. In the remain-
ing cases, the required statement has been checked with the help of the computer pack-
age COCO2P [9].
6 Concluding remarks and open problems
6.1 Pseudocyclic schemes
A scheme is said to be k-equivalenced (and just equivalenced if k is irrelevant) if all irreflex-
ive basis relations of it have valency k. It is known that every k-equivalenced scheme is
pseudocyclic for 1 ≤ k ≤ 4; this follows from results obtained in [12,14,15] and [13, The-
orem 3.1].
By Theorem 1.1, if X is a schurian equivalenced scheme of sufficiently large degree,
which is not trivial or Frobenius, then either X is exceptional, i.e., the Hollmann or Passman
scheme, or the inclusion Aut(X ) ≤ AΓL(1, q) holds for some q. In all cases except for the
last one, X is pseudocyclic, see Propositions 3.2, 5.2(2), and 5.4(2), and [13, Theorem 3.1].
In the latter case, the group AΓL(1, q) can contain 32 -transitive subgroups which are not 2-
equivalent to Frobenius groups (such a subgroup is always primitive). We do not know
whether the scheme of at least one of these subgroups is not pseudocyclic.
6.2 Separability number
Finding the exact values of s(X ) for an exceptional scheme X is still an open problem. A
direct calculation shows that these schemes are separable for small q.
6.3 Superschurian schemes
The following concept was first formulated many years ago in discussions of the third
author with Sergei Evdokimov. A scheme X is said to be superschurian if the extension of
X with respect to every set of points is schurian. In particular, all superschurian schemes are
schurian. In fact, only a few families of superschurian schemes are known; these include
partly regular schemes, cyclotomic schemes over finite fields, normal circulant schemes,
and some TI-schemes [3, 5, 13]. Theorem 3.7 implies that any large Hollmann scheme is
superschurian. We do not know whether other exceptional schemes are superschurian.
6.4 Base number
The base number b(X ) of a coherent configuration X is defined to be the smallest number
of points such that the extension of X with respect to them is the discrete configuration,
see [2, Section 3.3.2]. In general, the base number of the group Aut(X ) is less than or
equal to b(X ). The equality is attained, for example, if X is a partly regular coherent
configuration. By virtue of this observation, Theorems 3.7, 5.3, and 5.5 imply that except,
G. Chen et al.: A characterization of exceptional pseudocyclic association schemes 149
possibly, for several small Hollmann schemes the equality holds also for all exceptional
pseudocyclic schemes.
In fact, the base number of an exceptional scheme of enough large degree is bounded by
3. This fact can be used to construct a polynomial-time algorithm recognizing whether or
not a given scheme is exceptional. Taking the above discussion into account, this reduces
the recognition problem for the class of schurian equivalenced schemes (see [18, p.281]) to
the non-Frobenius schemes X for which Aut(X ) ≤ AΓL(1, q).
6.5 Bound in Theorem 4.1
Denote by f(n) the maximum of the ratio nc(X )k(X ) taken over all non-partly-regular co-
herent configurations X of degree n, where k(X ) is the maximal cardinality of a fiber
of X . Clearly, f(n) > 0. Theorem 4.1 states that f(n) < 2. It would be interesting to
find the function f(n) explicitly. We have found a (schurian) non-partly-regular coherent
configuration with parameters
n = 24, k = 8, c = 4,
which shows that f(24) ≥ 3/4.
ORCID iDs
Gang Chen https://orcid.org/0000-0001-5173-6710
Jiawei He https://orcid.org/0000-0001-7526-1246
Ilia Ponomarenko https://orcid.org/0000-0003-2444-731X
Andrey Vasil’ev https://orcid.org/0000-0002-6498-0138
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