Modular Representations in Coherent Configurations

• Modular representation theory studies the semisimplicity and structure of Bose-Mesner algebras in coherent configurations, with emphasis on field characteristics and combinatorial invariants.
• Frame numbers, p-ranks, and Jacobson radical criteria are central to determining when adjacency and design algebras are semisimple over F_p.
• Analysis of rank-3 schemes from partial geometries and type [3,2;3] designs provides insights into ongoing challenges such as Wedderburn decompositions and Gabriel quiver determinations.

Modular representation theory of coherent configurations, particularly those arising from partial geometries and their generalizations to designs of type [3,2;3], centers on the structure and semisimplicity of adjacency (or Bose-Mesner) algebras over fields of positive characteristic. Shimabukuro's work provides a comprehensive treatment of frame numbers, the Jacobson radical, $p$-ranks, and the links between the point-scheme algebras and the more intricate design algebras associated with coherent configurations (Shimabukuro, 6 Dec 2025).

1. Rank-3 Association Schemes from Partial Geometries

Partial geometries $\mathrm{pg}(s,t,\alpha)$ are incidence structures $(P, \mathcal{L})$ characterized by:

• Each line is incident with $s+1$ points;
• Each point lies on $t+1$ lines;
• Any two points are joined by at most one line;
• For any point $x \notin L$, there are exactly $\alpha$ lines through $x$ meeting $L$.

The number of points is $v=(s+1)(st+\alpha)/\alpha$, the number of lines is $b=(t+1)(st+\alpha)/\alpha$, and the point-graph $\Gamma$ is strongly regular with parameters

$$(v,\, k=s(t+1),\, \lambda=(s-1)+t(\alpha-1),\, \mu=\alpha(t+1)).$$

The (rank-3) association scheme $X=(P, \{R_0, R_1, R_2\})$ is formed with $R_0$ the diagonal, $R_1$ adjacency of $\Gamma$, $R_2$ the complement. The adjacency matrix $A$ of $R_1$ has eigenvalues $k=s(t+1)$, $r=s-\alpha$, $s'=- (t+1)$, with multiplicities $1$, $f$, $g$, respectively. These correspond to the three primitive idempotents of the adjacency algebra.

2. Frame Number and Semisimplicity Criteria

The Frame number $F_\mathrm{AS}(X)$ of a rank-3 commutative association scheme is

$F_\mathrm{AS}(X) = v^3 \frac{k(v-1-k)}{f g}.$

For schemes from $\mathrm{pg}(s,t,\alpha)$, explicit eigen-multiplicity calculations yield the closed formula: $$F_\mathrm{AS}(X) = v^2 \cdot (s + t + 1 - \alpha)^2.$$ This formula shows that the only possible prime divisors of $F_\mathrm{AS}(X)$ are those dividing either $v$ or $s+t+1-\alpha$. The critical consequence is the semisimplicity criterion: over a field $F_p$ of characteristic $p$, the adjacency algebra $F_p[X]$ is semisimple if and only if $p \nmid v$ and $p \nmid (s + t + 1 - \alpha)$. Thus, the locations and nature of modular (nonsemisimple) phenomena are entirely determined by these combinatorial invariants.

3. Jacobson Radical in Arithmetic Cases

The explicit structure of the Jacobson radical is determined by the divisibility relations between $p$, $v$, and $s+t+1-\alpha$. Defining $J$ as the all-ones matrix and $B=(A-kI)(A-rI)$, four mutually exclusive cases arise:

• (SS) $p\nmid v$, $p\nmid (s+t+1-\alpha)$: $F_p[X]$ is semisimple, $\operatorname{Rad}(F_p[X]) = 0$.
• (V) $p\mid v$, $p\nmid (s+t+1-\alpha)$: $\operatorname{Rad}(F_p[X]) = F_p \cdot J$, with $\dim=1$.
• (R) $p\nmid v$, $p\mid (s+t+1-\alpha)$: $\operatorname{Rad}(F_p[X]) = F_p \cdot B$, with $\dim=1$.
• (VR) $p\mid v$, $p\mid (s+t+1-\alpha)$: $$\operatorname{Rad}(F_p[X]) = F_p \cdot J \oplus F_p \cdot B$$ with $\dim=2$.

Algebraic properties such as $J^2 = vJ \equiv 0 \bmod p$ and $B^2 = 0$ in cases (R) and (VR) hold. These results fully classify the radical structure in all modular situations for rank-3 schemes from partial geometries [(Shimabukuro, 6 Dec 2025), Thm 3.10].

4. $p$-Ranks and Eigenvalue Criteria

Rank properties over $F_p$ hinge on the reduction of the spectrum modulo $p$. The characteristic polynomial of $A$ splits as $(x-k)(x-r)^f(x-s')^g$. If $p$ does not divide the pairwise differences of the eigenvalues, $A$ is diagonalizable over the algebraic closure of $F_p$ and the $p$-rank is determined by the vanishing of any eigenvalue modulo $p$:

• All eigenvalues $\not\equiv 0$: $\operatorname{rank}_p(A)=v$.
• $k\equiv 0$: $\operatorname{rank}_p(A) = v-1$.
• $r\equiv 0$: $\operatorname{rank}_p(A) = v-f$.
• $s' \equiv 0$: $\operatorname{rank}_p(A) = v-g$.

For generic $p$ not dividing key combinatorial data, one has full rank. This computation governs, for example, the kernel structure of the adjacency matrices and the dimension of the radical as related to $p$.

5. Coherent Configurations of Type [3,2;3] and Design Algebras

Strongly regular designs $(P, B, F)$ yield coherent configurations $X$ of type [3,2;3] with $P\uplus B$ as the point set and ten relations: three on $P$, three on $B$, and four mixed. The adjacency algebra over $\mathbb{C}$ decomposes as

$$\mathbb{C}[X] \cong \mathbb{C} \oplus \mathbb{C} \oplus M_2(\mathbb{C}) \oplus M_2(\mathbb{C})$$

(two trivial 1-dimensional and two nontrivial 2-by-2 blocks). Sharafdini's extension gives a Frame number $F_{\mathrm{CC}}(X)$ such that $F_p[X]$ is semisimple if and only if $p\nmid F_{\mathrm{CC}}(X)$. The point-scheme algebra $F_p[P]$ always embeds in the design algebra $F_p[X]$. Consequently, if $F_p[P]$ is not semisimple, neither is $F_p[X]$, and the radical of the point-scheme algebra injects as a direct summand into the radical of the design algebra.

In characteristic $2$, it is shown that $\operatorname{Comm}(\mathbb{C}[X])$ has rank 4, limiting the number of simple factors. The point-side radical injects, and computer calculations establish the existence of additional independent nilpotent elements on the design side, increasing the radical dimension to at least 2.

6. The Case of $\mathrm{GQ}(2,2)$ and Open Problems

For the unique generalized quadrangle $\mathrm{GQ}(2,2) = \mathrm{pg}(2,2,1)$, parameters are $(s,t,\alpha)=(2,2,1)$, $v=15$, $s+t+1-\alpha=4$. The Frame number is $15^2 \cdot 4^2 = 3600=2^4 \cdot 3^2 \cdot 5^2$, so the nonsplit cases are precisely $p \in \{2,3,5\}$.

Specialization of the general theory yields:

• For $p=3,5$: case (V), $\operatorname{Rad}(F_p[P]) = F_p\cdot J_P$, $\dim=1$.
• For $p=2$: case (R), $\operatorname{Rad}(F_2[P])=F_2\cdot B$, with $B=(A-6I)(A-1I)$ and $\dim=1$.
• $2$-rank of $A$ is $14$ (1-dimensional kernel given by all-ones vector).

On the design side of the associated type [3,2;3] configuration, the radical strictly contains the point-side radical, with computer experiments showing

• $\dim\,\operatorname{Rad}(F_2[X])=4$,
• $\dim\,\operatorname{Rad}(F_2[X])^2=2$,
• Loewy length at least $3$ ($\operatorname{Rad}^2 \neq 0$, $\operatorname{Rad}^3=0$).

The semisimple quotient $F_2[X]/\operatorname{Rad}$ has dimension $6$ (out of $10$), with at most $4$ simple blocks, but the precise structure – including the explicit Wedderburn decomposition and Gabriel quiver – remains open (Problem~6.8 of (Shimabukuro, 6 Dec 2025)).

7. Context, Significance, and Open Directions

The modular theory for coherent configurations of rank 3 from partial geometries is now essentially explicit except in situations requiring detailed analysis of mixed radicals and projective modules for larger coherent configurations. The extension to design algebras of type [3,2;3] and, specifically, the full modular representation structure for $\mathrm{GQ}(2,2)$ over $F_2$ remain open. Resolving the Wedderburn decomposition and computing the Gabriel quiver for these highly structured algebras is a challenging open problem with potential ramifications for the understanding of modular representations of finite incidence geometries and their automorphism groups.

For full technical details, explicit formulas, and further context, see Shimabukuro, "Frame Numbers and Jacobson Radicals for Partial Geometries and Related Coherent Configurations" (Shimabukuro, 6 Dec 2025), especially Sections 3–6.