Twisted primitive group association schemes

Abstract: We give results on the question of whether the intersection numbers of a primitive group association scheme determine it up to combinatorial isomorphism. For $G=\operatorname{PSL}(2,q)$, where $q$ is an odd prime power with $q=11$ or $q\ge 17$, or $q=2<sup>f$ with $f\ge3$, we construct a Schur partition that is algebraically isomorphic to the partition of $G$ into conjugacy classes but not combinatorially isomorphic to it. Consequently, the corresponding primitive group association schemes are not determined up to combinatorial isomorphism by their intersection numbers; in particular, they are non-separable. For $\mathfrak A_6$ and $\mathfrak A_8$, we also explicitly construct Schur partitions that are algebraically isomorphic to the corresponding partitions into conjugacy classes but not combinatorially isomorphic to them.

• The paper constructs an infinite family of algebraically isomorphic but combinatorially non-isomorphic, non-Schurian Cayley schemes from PSL(2,q) by switching square-class cosets in a Borel subgroup, for q=11 or q≥17 odd and q=2^f with f≥3.
• Twisted primitive group association schemes use local-graph invariants, clique structure, connectivity, and spectral conditions to prove non-isomorphism, while explicit examples for A6 and A8 demonstrate the method beyond PSL(2,q).
• An exhaustive survey of 6,065 groups of order at most 200 found 256 nontrivial twisted Schur partitions across 48 group types, showing that primitivity does not ensure separability or determination by intersection numbers.

Background and motivation

A recurring question in finite group theory asks how much of a finite group is encoded by its ordinary character table. The group association scheme $\mathfrak X(G)$, built from the partition of $G$ into conjugacy classes, has as its first eigenmatrix exactly the character table of $G$ (up to normalization), so the analogous combinatorial question is whether the intersection numbers of $\mathfrak X(G)$ determine it up to combinatorial isomorphism. Positive results exist for $\mathfrak A_5$, $PSL(2,7)$, and symmetric groups [Tomiyama; Tomiyama–Yamazaki], while Yoshiara and Terada constructed negative examples using split and non-split extensions of non-simple groups. Since closed subsets of $\mathfrak X(G)$ correspond to normal subgroups of $G$, primitivity of $\mathfrak X(G)$ is equivalent to simplicity of $G$; prior to this work, no group association scheme of a non-abelian finite simple group was known to fail determinacy by intersection numbers, even among Cayley association schemes over the same group. The paper supplies such examples uniformly across an infinite family.

The twisting construction for $G$0

Let $G$1 be a prime power and $G$2. For $G$3, the map $G$4 permutes the square classes. The construction modifies only the split semisimple conjugacy classes inside a fixed Borel subgroup $G$5: writing $G$6 for the intersections with the cosets $G$7 of the Sylow $G$8-subgroup structure inside $G$9, each split class $G$0 is replaced by

$G$1

with all other classes left unchanged. The main theorem states that $G$2 is a Schur partition algebraically isomorphic to the conjugacy-class partition, and that if $G$3 it is not combinatorially isomorphic to it.

The proof of algebraic isomorphism rests on two counting facts: products within the Borel subgroup satisfy $G$4 with unique factorization counts, and for $G$5 the distribution of traces of $G$6 as $G$7 ranges over $G$8 is independent of $G$9 — in odd characteristic because squaring a nonconstant affine function yields a fixed distribution over squares, and for the two unipotent classes via a parity argument showing that replacing $\mathfrak X(G)$0 by $\mathfrak X(G)$1 preserves the relevant square class when $\mathfrak X(G)$2 is a square.

Combinatorial non-isomorphism is established through local graphs. A key lemma shows that, for $\mathfrak X(G)$3, cliques of size $\mathfrak X(G)$4 in the unipotent Cayley graph are precisely left cosets of Sylow $\mathfrak X(G)$5-subgroups — proved by trace computations bounding mixed-Sylow cliques to size at most $\mathfrak X(G)$6 (or $\mathfrak X(G)$7 in characteristic $\mathfrak X(G)$8). In the switched basic set $\mathfrak X(G)$9, elements of $\mathfrak A_5$0 lie in $\mathfrak A_5$1-cliques while elements outside $\mathfrak A_5$2 do not, so the local graph $\mathfrak A_5$3 is not vertex-transitive. Since vertex-transitivity of local graphs is preserved under combinatorial isomorphism, the schemes cannot be combinatorially isomorphic. The same non-vertex-transitivity also implies the twisted Cayley scheme is non-Schurian.

The excluded parameters are exactly those with $\mathfrak A_5$4, i.e., $\mathfrak A_5$5: odd characteristic $\mathfrak A_5$6 and even characteristic $\mathfrak A_5$7. Consequently, for every odd prime power $\mathfrak A_5$8 or $\mathfrak A_5$9, and every $PSL(2,7)$0 with $PSL(2,7)$1, the primitive group association scheme $PSL(2,7)$2 admits an algebraically isomorphic but combinatorially non-isomorphic Cayley scheme over the same group; in particular it is non-separable. This is the first infinite family of primitive group association schemes from finite simple groups with this failure.

Twisted conjugacy classes: a general criterion

A second, independent mechanism replaces two conjugacy classes $PSL(2,7)$3 of equal size, inverse-closed as a pair, by a new partition $PSL(2,7)$4 of their union. Writing $PSL(2,7)$5 and $PSL(2,7)$6, the hypotheses require that $PSL(2,7)$7 be a simultaneous eigenvector for all other class sums, that $PSL(2,7)$8 lie in the span of the remaining class sums together with $PSL(2,7)$9, and analogously for $\mathfrak X(G)$0 with $\mathfrak X(G)$1. Under these conditions the resulting family is a Schur partition whose multiplication table coincides with that of the original classes, yielding an explicit algebraic isomorphism fixing all other basic sets.

A rigidity criterion follows from spectral considerations: any admissible sign vector $\mathfrak X(G)$2 lies in the same eigenspaces of the adjacency matrices $\mathfrak X(G)$3 and $\mathfrak X(G)$4 as the original $\mathfrak X(G)$5. Hence if any relevant eigenspace is one-dimensional, no nontrivial repartition exists. This criterion drives both the negative results for alternating groups and the computational classification below.

Twists for alternating groups

For $\mathfrak X(G)$6, the two split $\mathfrak X(G)$7-cycle classes $\mathfrak X(G)$8 (each of size $\mathfrak X(G)$9) satisfy the hypotheses, with eigenvalues $G$0 against the other class sums and $G$1. The new parts are defined via the number of edges of a $G$2-cycle crossing the cut $G$3: each part combines the crossing-number-$G$4 elements of one split class with the crossing-number-$G$5 elements of the other, equivalently the union of nonidentity elements of eighteen Sylow $G$6-subgroups. The distinguishing invariant is the local graph $G$7, which decomposes into six copies of the icosahedral graph, whereas all corresponding local graphs of the twist are connected $G$8-regular graphs on $G$9 vertices. Triangle counts ($\mathfrak X(G)$0 vertices with value $\mathfrak X(G)$1, $\mathfrak X(G)$2 with value $\mathfrak X(G)$3) show the twist is non-Schurian. A computer check confirms these ten cut-based partitions exhaust the possibilities and yield a single isomorphism type. Notably, although $\mathfrak X(G)$4, this twist is not an instance of the Borel-switching construction, since $\mathfrak X(G)$5 excludes $\mathfrak X(G)$6 there.

For $\mathfrak X(G)$7, the two split $\mathfrak X(G)$8-classes (each of size $\mathfrak X(G)$9) are repartitioned cell-by-cell over the $G$0 cyclic subgroups of order $G$1: within each cell, the quadratic-residue and nonresidue thirds are reassigned according to orientations transported along seven orbits of a subgroup $G$2 acting on cells. The result gives $G$3 with $G$4, verified spectrally to satisfy the hypotheses. Non-isomorphism follows from component counts: $G$5 has eight connected components (indexed by the fixed point), while $G$6 and $G$7 are connected. The twist is again non-Schurian, with triangle counts $G$8 and $G$9 both occurring.

By contrast, for $G$00 the relevant eigenspaces are one-dimensional (eigenvalues $G$01 for $G$02 on $G$03; $G$04, $G$05, $G$06, and $G$07 respectively for the $G$08- or $G$09-class graphs), so the rigidity criterion forces every admissible partition to be the original one. These groups are rigid with respect to this method, though for $G$10 and $G$11 uniqueness was already known by stronger classifications.

Computational survey of small groups

An exhaustive GAP search over all $G$12 group isomorphism classes of order at most $G$13 finds that $G$14 have at least one pair of conjugacy classes satisfying the hypotheses, and $G$15 nontrivial unordered partitions satisfy the conditions of the general lemma. Of these, $G$16 yield Schur partitions not combinatorially isomorphic to the conjugacy-class partition, occurring across $G$17 group isomorphism types of orders $G$18 and $G$19; the smallest example is $G$20. For $G$21, three non-original partitions exist but all produce schemes combinatorially isomorphic to $G$22, consistent with Tomiyama's classification, in which the other two isomorphism types have no regular subgroup in their automorphism groups.

Limitations and open questions

Several boundaries of the constructions are acknowledged explicitly. The Borel-switching method fails precisely when $G$23, leaving $G$24 as the only simple case in the $G$25 family where the question remains open. The general repartitioning lemma does not apply to $G$26 or $G$27, though rigidity relative to this particular method does not preclude twists obtainable by other means. The count of $G$28 partitions does not quotient by combinatorial isomorphism of the resulting schemes, so the number of genuinely distinct twisted schemes among them is not determined. The paper leaves three questions open: which non-abelian finite simple groups $G$29 have $G$30 undetermined by its intersection numbers; whether $G$31 is uniquely determined, possibly via a different switching construction; and whether further inequivalent twists exist for $G$32, $G$33, and $G$34, including algebraically isomorphic schemes that are not Cayley schemes.

Conclusion

The paper establishes that primitivity alone does not force separability of group association schemes: for all but finitely many small parameters, $G$35 shares its intersection numbers with a non-isomorphic, non-Schurian Cayley scheme obtained by switching square-class cosets inside a Borel subgroup. Explicit twists for $G$36 and $G$37, a general repartitioning criterion with a spectral rigidity converse, and a systematic small-group survey round out the picture, delineating sharply where the intersection numbers of a group association scheme cease to determine its combinatorial structure.