Almost Commutative Terwilliger Algebra

• The paper presents the complete classification and characterization of almost commutative Terwilliger algebras, elucidating their unique one-dimensional non-primary modules.
• It provides an explicit block decomposition via Wedderburn structure, employing Kronecker products and Fourier inversion to determine primitive idempotents.
• Results extend to group association schemes from abelian, Frobenius, and Camina groups, offering insights for combinatorial regularity and spectral analysis.

An almost commutative Terwilliger algebra is a semisimple subalgebra of a matrix algebra constructed from an association scheme, defined by the property that all non-primary irreducible modules are one-dimensional. This structure appears most notably in group association schemes derived from conjugacy classes of finite groups, as well as in certain association schemes arising from combinatorial and geometric constructions. Rigorous criteria established by Rie Tanaka and subsequent research (Bastian et al., 13 Sep 2024, Bastian et al., 19 Sep 2025) yield complete characterizations and classification theorems for almost commutative Terwilliger algebras, alongside explicit block decompositions, dimension formulas, and a connection to combinatorial regularity.

1. Formal Characterization and Equivalent Conditions

For a commutative association scheme $\mathcal{A} = (\Omega, \{A_0, ..., A_d\})$ and basepoint $x \in \Omega$, the Terwilliger algebra $T(x)$ is generated by the Bose–Mesner algebra and dual Bose–Mesner algebra (i.e., the diagonal matrices $E^*_i(x)$ marking relations from $x$). Tanaka’s definition of almost commutative (AC) is that $T(x)$ is semisimple and every non-primary irreducible $T(x)$-module is one-dimensional (Bastian et al., 13 Sep 2024, Bastian et al., 19 Sep 2025). The following conditions are equivalent:

• (a) For some/all $x \in \Omega$, all non-primary irreducible modules are one-dimensional.
• (b) The intersection numbers satisfy: for any $h \neq i$, there exists exactly one $j$ with $p_{ij}^h \neq 0$; that is, the product $A_i A_j$ "lands" inside a single relation.
• (c) Analogous condition for the Krein parameters of the dual Bose–Mesner algebra.
• (d) The scheme decomposes as a wreath product of one-class schemes and abelian group schemes. A sixth equivalent condition for schemes coming from a commutative Schur ring is that, for given $x \in P_i$, $y \in P_j$ (principal sets), and $xy \in P_h$, if $P_i \neq P_j^*$, then $P_i P_j = P_h$ (Bastian et al., 19 Sep 2025). For group association schemes, this sharply restricts possible group structures.

2. Classification of Underlying Groups and Association Schemes

The only finite groups whose group association scheme yields an AC Terwilliger algebra are:

• Finite abelian groups.
• Frobenius groups of the form $\mathbb{Z}_p^r \rtimes \mathbb{Z}_{p^r−1}$ (for prime $p$, $r > 0$).
• Non-abelian Camina $p$-groups, including certain nilpotency class 2 or 3 groups where every non-central conjugacy class is a coset of the derived subgroup.
• The group $\mathbb{Z}_3^2 \rtimes Q_8$ (Bastian et al., 13 Sep 2024, Bastian et al., 19 Sep 2025). These classifications rely crucially on the coset structure of nontrivial conjugacy classes and the uniqueness of intersection parameters. For association schemes from strong Gelfand pairs $(G, H)$, only abelian $G$ or Frobenius groups with cyclic complement and kernel equal to $G'$, where $G'$ is abelian or a Camina $p$-group, produce such algebras (Bastian et al., 19 Sep 2025).

3. Block Structure and Primitive Idempotents

The Wedderburn decomposition of an AC Terwilliger algebra has one primary component (dimension $(d+1)^2$ for a $d$-class scheme) and all remaining summands are one-dimensional ideals. For $G = \mathbb{Z}_p^r \rtimes \mathbb{Z}_{p^r−1}$, dimension is $p^{2r} + p^r-1$. For abelian $G$, dimension is $|G|^2$ (Bastian et al., 13 Sep 2024, Bastian et al., 19 Sep 2025). Each non-primary primitive idempotent is explicitly described: for block corresponding to a conjugacy class $C_i$,

$$B_i = \frac{-1}{|C_i| - 1} J_{|C_i|} + \left(1 - \frac{-1}{|C_i| - 1}\right) I_{|C_i|},$$

where $J_{|C_i|}$ is the all-ones matrix, $I_{|C_i|}$ the identity. For Camina $p$-groups, idempotents are constructed from Fourier sums over character values, often involving Kronecker products exploiting the cyclic structure of the center and factor group (Bastian, 14 Sep 2024).

4. Structural Decomposition: Wreath Products and Schur Rings

A salient feature is the decomposition of the group association scheme as a wreath product of simpler association schemes. For Frobenius groups:

$$\mathcal{G}(G) = \mathcal{K}_{p^r} \wr \mathcal{G}(\mathbb{Z}_{p^r−1}),$$

where $\mathcal{K}_{p^r}$ is the trivial one-class scheme (Bastian et al., 19 Sep 2025, Bastian et al., 19 Sep 2025). For Camina groups:

$$\mathcal{G}(G) = \mathcal{G}(Z(G)) \wr \mathcal{G}(G/Z(G)), \quad \text{(class 2)}$$

$$\mathcal{G}(G) = \mathcal{G}(Z(G)) \wr \mathcal{G}(G'/Z(G)) \wr \mathcal{G}(G/G'), \quad \text{(class 3)}$$

These decompositions are algebraically realized as Kronecker products in the adjacency matrices and module structure.

5. Relation to "Thinness" and Commutativity

The almost commutative property is reflected in the "thinness" of the module structure: apart from one large irreducible representation (the primary module), every other is one-dimensional (Bastian et al., 13 Sep 2024). The algebra's center is large; the regular structure enforces commutativity within blocks and trivial mixing between different non-primary summands. In terms of intersection numbers, for any $h \neq i$, there is at most one $j$ such that $p_{ij}^h \neq 0$; consequently, products of non-central classes are sharply localized, yielding a highly structured algebraic setting.

6. Applications and Interpretative Significance

These results delineate precisely which group-theoretic and combinatorial structures admit almost commutative Terwilliger algebras—implying that spectral and module-theoretic analyses (such as spectral graph theory, coding theory, and combinatorial optimization) in these settings are significantly simplified due to the block and thin structure. The explicit formulas, decomposition via wreath products, and primitive idempotent construction facilitate practical computations and representation classification. The framework applies to analysis in algebraic combinatorics, symmetry studies in quantum theory, and design of experiments via association schemes.

7. Further Developments and Extensions

The construction methods—such as the use of Kronecker product decompositions and Fourier inversion for idempotent analysis—have broad implications for explicit Wedderburn decompositions in related algebraic structures. Recent research extends these classification results to association schemes derived from strong Gelfand pairs and investigates cellular and homological properties in quasi-thin association schemes (Chen et al., 28 Oct 2024). The concept of almost commutativity is further linked to triply regularity and spectral triple constructions in noncommutative geometry (Aastrup et al., 26 Mar 2025), suggesting potential bridges between algebraic combinatorics and mathematical physics.

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In summary, the theory of almost commutative Terwilliger algebras provides a unified framework for identifying and analyzing highly structured semisimple subalgebras in association schemes. The combination of group-theoretic classification, combinatorial regularity, explicit block decomposition, and connection to wreath products enables complete characterization and practical computation of their algebraic and representation-theoretic properties.