Primitive Central Idempotents

• Primitive central idempotents are idempotent elements in group algebras that are both indivisible and central, serving as atomic units in the Wedderburn decomposition.
• They are explicitly constructed using subgroup data and strong Shoda pairs to yield minimal two-sided ideals in rational group algebras.
• Their determination facilitates precise module and representation analysis by isolating simple components and constructing full matrix units.

A primitive central idempotent is an idempotent element—which is both primitive and central—in the algebraic structure under consideration, such as a group algebra, where it plays a key role in the Wedderburn-Artin decomposition, representation theory, and explicit module constructions. In rational group algebras of finite nilpotent or strongly monomial groups, as well as in other algebraic settings, primitive central idempotents serve as the atomic decomposition units of the algebra, yielding minimal two-sided ideals and distinguishing simple components up to isomorphism. The theory of primitive central idempotents in rational group algebras is characterized by explicit, often character-free constructions that exploit subgroup-theoretic data such as strong Shoda pairs, cyclic subgroups, and actions of normalizers.

1. Definitions and Foundational Properties

Let $A$ be a (possibly noncommutative) unital ring. An idempotent is $e\in A$ with $e^2=e$. An idempotent $e$ is central if it commutes with all elements of $A$, i.e., $e x = x e$ for every $x\in A$. An idempotent is primitive if it cannot be written as the sum of two nonzero orthogonal idempotents (i.e., if $e=f+g$ with $f,g$ idempotent and $fg=gf=0$, then either $f=0$ or $g=0$). In semisimple algebras, the set of primitive central idempotents forms a complete set of pairwise orthogonal idempotents that sum to the identity, each acting as the identity on exactly one simple component.

In group algebras $\mathbb{Q}G$ for a finite group $G$, primitive central idempotents correspond to the simple components in the Wedderburn decomposition: $\mathbb{Q}G \cong \bigoplus_i M_{n_i}(D_i)$ where each summand is a matrix algebra over a division algebra $D_i$ whose center is a number field. Here, each primitive central idempotent "cuts out" one such simple component.

2. Character-Free Constructions: Shoda Pairs and their Generalizations

For finite nilpotent and strongly monomial groups, a principal method of producing all primitive central idempotents in $\mathbb{Q}G$ is via explicit, character-free constructions built from subgroup pairs.

A Shoda pair $(H,K)$ consists of subgroups $K \leq H \leq G$ with $K \unlhd H$ and $H/K$ cyclic, such that a linear character of $H$ with kernel $K$ induces an irreducible character of $G$. A strong Shoda pair further requires orthogonality of the corresponding idempotent's conjugates and normalizer containment ($H\leq N_G(K)$), ensuring that the sum over conjugates produces a central idempotent that is also primitive.

Given such a strong Shoda pair, the construction proceeds via the following steps (Jespers et al., 2010, Jespers et al., 2012, Bakshi et al., 13 Jan 2024):

1. Let $\widehat{X} = \frac{1}{|X|}\sum_{x\in X} x$ be the “averaging idempotent” for a subgroup $X\leq H$.
2. Define $$\varepsilon(H,K) = \prod_M (\widehat{K} - \widehat{M})$$, taking the product over all minimal normal subgroups $M$ of $H$ properly containing $K$.
3. Form the primitive central idempotent as

$e(G, H, K) = \sum_{t\in T} \varepsilon(H, K)^{t}$

where $T$ is a set of representatives for the cosets of the centralizer of $\varepsilon(H, K)$ in $G$.

1. The set of all such $e(G, H, K)$, as $(H, K)$ runs over a complete set of strong Shoda pairs (possibly generalized), gives all primitive central idempotents in $\mathbb{Q}G$.

This "character-free" approach reduces dependency on explicit knowledge of character tables, instead leveraging subgroup lattice data, normalizer actions, and explicit element manipulations. In the case of generalized strongly monomial groups, the construction is further refined by tracking inductive chains of subgroups and crossed product structures, allowing recursive decomposition of simple components into explicit matrix algebra forms (Bakshi et al., 13 Jan 2024).

3. Relation to Irreducible Characters and Cyclotomic Fields

In full generality, the classical construction of primitive central idempotents in a group algebra over a splitting field (e.g., $\mathbb{C}G$) uses irreducible characters: $$e(\chi) = \frac{\chi(1)}{|G|} \sum_{g\in G} \chi(g^{-1})g$$ for each irreducible character $\chi$. Over $\mathbb{Q}$, Galois descent is required because $\mathbb{Q}$ is not always a splitting field for $G$: $$e_{\mathbb{Q}}(\chi) = \sum_{\sigma\in \mathrm{Gal}(K/\mathbb{Q})} \sigma(e(\chi))$$ where $K$ is the field of values of $\chi$, and the sum is over the Galois orbit of $\chi$. For certain group classes, explicit formulas relate these to Shoda pair constructions, and for abelian $G$, a further concrete form is available via cyclotomic fields. In this case,

$$\mathbb{Q}G \cong \bigoplus_{d\,|\,n} \mathbb{Q}(\zeta_d)^{m_d}$$

where $\zeta_d$ is a primitive $d$-th root of unity, and $m_d$ is the multiplicity determined by the number of elements or cyclic subgroups of given order (Kulkarni et al., 2015, Janssens, 2013).

More generally, Artin induction yields expressions of rational-valued characters as linear combinations of induced characters from cyclic subgroups, which then allows the primitive central idempotents to be written as $\mathbb{Q}$-linear combinations of building blocks associated to conjugacy classes of cyclic subgroups or their pairings (Janssens, 2013). Explicit formulas for such decompositions involve Möbius functions, group indices, and evaluations of characters at generators of cyclic subgroups.

4. Explicit Wedderburn Decomposition and Minimal Matrix Units

Once a primitive central idempotent $e$ is constructed—particularly for $\mathbb{Q}G$ in a nilpotent or strongly monomial group—one can describe the simple component $\mathbb{Q}G e$ as a full matrix algebra over a division algebra (cyclotomic field or its fixed field under a Galois group) via a crossed product structure: $\mathbb{Q}G e \cong M_r(D)$ where, under suitable Schur index and twisting conditions, $D$ is a field (Jespers et al., 2010, Jespers et al., 2012, Bakshi et al., 13 Jan 2024).

Within such a simple component, a complete set of pairwise orthogonal primitive idempotents, corresponding to diagonal matrix units, can be constructed explicitly. In the rational group algebra, these idempotents are obtained by conjugating the basic idempotent, often constructed from subgroup or coset representatives, suitably lifted via sequence of isomorphisms along strong inductive chains and combining with centralizer basis elements: $$\left\{\, t^{-1} z_i^{-1} a^{-1} e(H, K) a z_i t : t \in T,\, 1\leq i \leq k \,\right\}$$ with notation as above, forms the complete set in $\mathbb{Q}G e(H, K)$ (Bakshi et al., 13 Jan 2024).

For cyclic or abelian group algebras, explicit Lagrange interpolation or product factorization results in concise formulas for the matrix units and their corresponding projections—these coincide with evaluations at roots of unity and are central due to abelianity (Martínez et al., 2014, Kulkarni et al., 2015).

5. Structural and Module-Theoretic Consequences

The determination of primitive central idempotents has major implications:

• Wedderburn Decomposition: Each primitive central idempotent corresponds to a simple summand in the decomposition of $\mathbb{Q}G$. Their explicit determination enables full description of the algebra's structure (Jespers et al., 2010, Jespers et al., 2012, Kulkarni et al., 2015).
• Analysis and Construction of Units: Knowledge of the matrix units (built from primitive central idempotents) enables the explicit construction of subgroups of finite index in the unit group of the integral group ring $\mathbb{Z}G$ and the construction of free non-abelian subgroups using pairs of elementary units tied to the idempotents (Jespers et al., 2010, Jespers et al., 2012).
• Module and Representation Theory: Primitive central idempotents project onto minimal two-sided ideals, allowing the isolation and analysis of simple modules (irreducible representations). These idempotents act as identity elements in their respective simple component.
• Algorithmic and Computational Aspects: Explicit formulas for primitive central idempotents enable their computation in practical settings such as group cohomology, code construction, or computational group theory environments (e.g., GAP), especially for complex or large groups (Janssens, 2013, Bakshi et al., 13 Jan 2024, Martínez et al., 2014).

6. Applicability and Broader Context

While the constructions above are most completely understood for finite nilpotent, abelian-by-supersolvable, and (generalized) strongly monomial groups, extensions and analogues exist (with additional complications) in more general settings:

• In Clifford and Brauer algebras, analogous primitive idempotents are described using non-group-theoretic data, e.g., via the action of certain group elements or via idempotency properties of diagram algebras (Ablamowicz et al., 2010, King et al., 2016).
• In semiring and ring-theoretic contexts, the presence of central idempotents and their orthogonal decompositions sharply influence structural results, e.g., commutativity or Boolean properties, depending on complement and nilpotency conditions (Chintala, 2023, Dolžan, 11 Apr 2024).
• In categorical, coding, and algebraic combinatorics settings (such as Burnside rings, double Burnside rings, trivial source algebras), primitive central idempotents correspond to blocks or module decompositions with interpretations via Hom-sets, representation rings, and bisets, constructed via analogous ghost or mark maps and character-theoretic module summations (Boltje et al., 2012, Barker, 2018).

7. Summary Table: Explicit Constructions in Prominent Settings

Setting / Algebra	Construction of Primitive Central Idempotents
Rational group algebra of finite nilpotent group ($\mathbb{Q}G$)	Sum over G-conjugates of explicit idempotents from strong Shoda pairs; character-free formulas using subgroup data (Jespers et al., 2010, Jespers et al., 2012, Bakshi et al., 13 Jan 2024)
Abelian group algebra ($\mathbb{Q}[A]$)	Product form from long presentations; compression into products of idempotents for each cyclic factor; identification with cyclotomic field projections (Kulkarni et al., 2015, Martínez et al., 2014)
General finite group	Rational linear combinations of basic elements attached to cyclic subgroup pairs, via Artin induction and Möbius function summations (Janssens, 2013)
Schur rings over cyclic groups	Lattice Schur ring idempotents built from normal subgroup lattices; explicit formulas using intersection and join (Misseldine, 2013)

• Rational group algebras of finite groups: from idempotents to units of integral group rings (Jespers et al., 2010)
• Group rings of finite strongly monomial groups: central units and primitive idempotents (Jespers et al., 2012)
• Primitive Central Idempotents of Rational Group Algebras (Janssens, 2013)
• Primitive Central idempotents and the Wedderburn Decomposition of a Rational Abelian Group Algebra (Kulkarni et al., 2015)
• Explicit idempotents of finite group algebras (Martínez et al., 2014)
• Rational group algebras of generalized strongly monomial groups: primitive idempotents and units (Bakshi et al., 13 Jan 2024)
• Primitive Idempotents of Schur Rings (Misseldine, 2013)
• Central idempotents of the bifree and left-free double Burnside ring (Boltje et al., 2012)
• On central idempotents in the Brauer algebra (King et al., 2016)
• Centrality and Partition of Idempotents (Chintala, 2023)
• Semirings generated by idempotents (Dolžan, 11 Apr 2024)