Wedderburn Decomposition of Rational Group Algebras

• Wedderburn decomposition expresses a rational group algebra as a direct sum of simple matrix algebras using strong Shoda pairs for explicit idempotent construction.
• The method details a character-free approach to compute primitive central idempotents and matrix units, simplifying the analysis of finite nilpotent groups.
• Its applications include determining unit groups of integral group rings and linking algebraic structure with representation-theoretic invariants.

A rational group algebra $\mathbb{Q}G$ of a finite group $G$ decomposes, by the Wedderburn–Artin theorem, as a direct sum of simple algebras, each isomorphic to a full matrix ring over a division algebra with center a finite extension of $\mathbb{Q}$. The explicit structure of this decomposition for various finite groups is central to understanding their representation theory, integral group rings, and arithmetic invariants. The rational group algebra of a finite nilpotent group offers a strikingly explicit and character‐free case, realized using the theory of strong Shoda pairs and central idempotents. This approach extends, with modifications, to broader classes of finite groups and yields applications in both constructing explicit elements in the group algebra and analyzing the associated unit group of the integral group ring.

1. Central Idempotent Decomposition via Strong Shoda Pairs

For a finite nilpotent group $G$, the primitive central idempotents of $\mathbb{Q}G$ are constructed without recourse to character theory using strong Shoda pairs $(H,K)$, where $K\leq H\leq G$, $H/K$ is cyclic, and additional normalizer conditions are satisfied. The explicit formula for each such idempotent is: $e(G,H,K) = \sum_{t \in T} t^{-1} E(H,K) t$ where $E(H,K)$ is an idempotent in $\mathbb{Q}H$ associated with the cyclic quotient $H/K$ and $T$ runs over a transversal of the centralizer of $E(H,K)$ in $G$ (Jespers et al., 2010). These idempotents are complete (parametrize all simple components), orthogonal, and constructed in a way that bypasses the need for the full character table.

For arbitrary finite groups, every primitive central idempotent of $\mathbb{Q}G$ is a $\mathbb{Q}$-linear combination of such $e(G,H,K)$, with $(H,K)$ strong Shoda pairs in subgroups of $G$. While the coefficients are not always explicit, this reduces the decomposition problem to computations structured by group-theoretic data.

2. Matrix Algebra and Crossed Product Structure of Simple Components

With the set of primitive central idempotents indexed by strong Shoda pairs available, the corresponding simple components are described as matrix algebras over crossed products of cyclotomic fields with finite groups: $$\mathbb{Q}G e(G,H,K) \cong M_{[G:N]}(\mathbb{Q}(\zeta_m) *^{\alpha}_{\tau} (N/H))$$ where:

• $m = [H:K]$; $\zeta_m$ is a primitive $m$th root of unity,
• $N = N_G(K)$ is the normalizer of $K$ in $G$,
• $[G:N]$ is the multiplicity (matrix size),
• the crossed product $\mathbb{Q}(\zeta_m) *^{\alpha}_{\tau} (N/H)$ encodes the action and possible twisting arising from nontrivial group extensions (Jespers et al., 2010).

This yields a direct sum decomposition: $$\mathbb{Q}G \cong \bigoplus_{(H,K)} M_{[G:N]}(\mathbb{Q}(\zeta_m) *^{\alpha}_{\tau} (N/H))$$ The algebraic structure of each simple factor is governed by the group structure (via $(H,K)$ and $N(H,K)$) and the associated cyclotomic extensions.

3. Explicit Construction of Matrix Units and Orthogonal Idempotents

Section 4 of (Jespers et al., 2010) details an algorithm for constructing a complete set of orthogonal primitive idempotents inside each Wedderburn component. Using appropriately chosen elements in the cyclotomic field and group-theoretic transversals, every matrix unit $E_{t,t'}$ in $\mathbb{Q}G e(G,H,K)$ can be written as: $E_{t,t'} = t^{-1} B_e t'$ where $t,t'$ range over a basis $T_e$ and $B_e$ is a fixed element from $N_G(K)$. These matrix units satisfy the full matrix algebra relations and allow for the explicit computation of isomorphisms between $\mathbb{Q}G e(G,H,K)$ and $M_n(D)$, where $D$ is the crossed product division algebra.

4. Schur Index Constraints and the Quaternion Case

A key structural result is that the Schur index of any simple component of $\mathbb{Q}G$ for nilpotent $G$ (and more generally, over fields of characteristic $0$) is at most $2$. If the Schur index is $2$, the corresponding component is a quaternion division algebra, occurring precisely when the Sylow $2$-subgroup of $G$ contains a quaternion section, a direct deduction from the strong Shoda pair description (Jespers et al., 2010). This captures and generalizes the classical result of Roquette on the Schur index for nilpotent group algebras.

5. Applications: Index Subgroups and Free Subgroups in Unit Groups

The constructive approach to the Wedderburn decomposition directly impacts the paper of the unit group $U(\mathbb{Z}G)$ of the integral group ring:

• Two nilpotent subgroups generated by matrix units (upper and lower triangular parts in each component) together with the central Bass units are shown to generate a subgroup of finite index in $U(\mathbb{Z}G)$ [(Jespers et al., 2010), Theorem 5.3].
• For components that are not division algebras, units of the form $1 + |G| \cdot E_{t,t'}$ (constructed from matrix units) are shown to generate free subgroups of rank $2$ in $U(\mathbb{Z}G)$.

The strong Shoda pair framework is essential for these constructions, as it gives explicit generators and allows for effective control over the algebra’s additive and multiplicative structure.

6. Extensions and Limitations Beyond Nilpotent Groups

For finite groups beyond the nilpotent case (such as some alternating groups or non-monomial groups), the explicit, character-free strong Shoda pair method does not yield a full set of idempotents or simple components, but every primitive central idempotent remains a rational linear combination of $e(G,H,K)$ from strong Shoda pairs in subgroups. While precise rational coefficients may not be readily determined, this establishes strong Shoda pairs as fundamental building blocks for all finite group algebras over $\mathbb{Q}$ (Jespers et al., 2010).

7. Impact and Broader Context

The explicit, character-free construction for nilpotent groups enables the computation of the full module and algebraic structure of $\mathbb{Q}G$, without appealing to advanced character theory or complex representation theory. This approach intersects with the paper of units of group rings, arithmetic applications, and algorithmic algebra (notably via effective descriptions of primitive idempotents and matrix units). It also provides a blueprint for analogous methods in broader settings (e.g., group algebras over number fields, integral group rings, and associated unit group problems) and links the algebraic decomposition to deep group-theoretic invariants.

The interpretation of the Wedderburn decomposition via strong Shoda pairs thus underpins both theoretical developments and explicit computation in the structure theory of rational group algebras and their integral analogues.