Primitive Idempotents in Minimal Ideals

• Primitive idempotents of minimal ideals are irreducible elements that generate minimal left, right, or two-sided ideals in various algebraic structures.
• They are constructed via methods in Clifford algebras, recursive techniques in monoids, and explicit character-theoretic formulas in group algebras.
• These idempotents reveal deep links between algebraic decompositions, symmetry groups, and computational applications in coding theory and categorical frameworks.

Primitive idempotents of minimal ideals are fundamental structural elements in a wide spectrum of algebraic systems, providing the irreducible building blocks for decompositions, representation theory, and module structure. Their role is especially prominent in settings where minimal one-sided ideals are characterized by their generation via primitive idempotents, and where the interplay with symmetries, stabilizers, and anti-involutions yields deep theoretical and practical ramifications.

1. Characterization and Construction in Clifford Algebras

In real Clifford algebras $\text{Cl}(V,Q)$ with signature $(p,q)$, a primitive idempotent $f$ is an element satisfying $f^2 = f$ and not expressible as the sum of two nonzero orthogonal idempotents. Such $f$ are systematically constructed as products of mutually commuting monomials $\beta_i$ with $\beta_i^2 = 1$: $$f = \tfrac{1}{2}(1 \pm \beta_1) \cdot \tfrac{1}{2}(1 \pm \beta_2) \cdots \tfrac{1}{2}(1 \pm \beta_k).$$ A complete set of $2^k$ mutually annihilating primitive idempotents $\{f_i\}$, derived via conjugation by monomials in the Salingaros vee group $G_{p,q}$, sums to the unity. Each $f_i$ generates a distinct minimal left ideal $S_i = \mathrm{Cl}_{p,q}f_i$, forming the spinor spaces parametrized by the signature. The structure of these idempotents directly encodes the underlying division ring $\mathbb{K}$ (where $\mathbb{K} = \mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$), and the classification of stabilizer subgroups $G_{p,q}(f)$ of $f$—normal in $G_{p,q}$ with explicit order and abelian/nonabelian nature depending on $(p-q) \bmod 8$—is essential for constructing spinor bases and for the analysis of module automorphisms.

The unique transposition anti-involution $T$ defined on $\mathrm{Cl}_{p,q}$ satisfies $T(\beta) = \beta^{-1}$ for basis monomials and leaves primitive idempotents fixed ($T(f) = f$). It induces on spinor representations a matrix conjugation: $$[T(u)] = \begin{cases} [u]^\mathrm{T} & (p-q) \equiv 0,1,2 \pmod{8} \ [u]^\dagger & (p-q) \equiv 3,7 \pmod{8} \ [u]^\ddagger & (p-q) \equiv 4,5,6 \pmod{8} \end{cases}$$ with real, complex, or quaternionic structure. The anti-involution further defines a new intrinsic spinor norm $\langle \psi, \phi \rangle := T(\psi)\phi$ that is invariant under a subgroup $G_{p,q}^\varepsilon$, and typically distinct from other classical norms ($\beta_+, \beta_-$) in the literature. The totality of these facts provides a robust mechanism for describing minimal ideals, their automorphism groups, and intrinsic pairings within Clifford module categories (1005.3558).

2. Recursive and Explicit Construction in Monoids and Group Algebras

R-Trivial Monoids

For finite R-trivial (and, by equivalence, weakly ordered) monoids, the collection of primitive idempotents $\{e_J\}_{J \in L}$ in the monoid algebra $\mathbb{C}S$ is recursively constructed via:

• Canonically defined "level" elements $T_J$ (products of idempotents at or below level $J$ in an associated finite semilattice $L$),
• Annihilators $A_J$ removing contributions from higher levels,
• Polynomials $P_J$ isolating desired terms,
• Recursive subtraction: $e_J = P_J (1 - \sum_{K\succ J} e_K)$.

This system yields pairwise orthogonal idempotents that project onto indecomposable projective modules, with applications including the $0$-Hecke algebra representations and left regular bands. The minimal left ideals $\mathbb{C}Se_J$ correspond to projective covers and are central to the computation of Cartan invariants and the structure of the quiver for $\mathbb{C}S$ (1009.4943).

Finite Group Algebras

For semisimple group algebras $FG$ of nilpotent groups $G$, the central idempotent associated to a simple component is constructed using strong Shoda pairs and cyclotomic classes. Primitive central idempotents are then refined to orthogonal primitive idempotents via explicit "factored" elements—e.g., by conjugating an explicitly built idempotent $\beta_{e_C}$ over a suitable transversal in $G$ relative to the stabilizer of a component. In matrix language, preimages of matrix units in the Wedderburn block yield a complete set of minimal one-sided ideals with explicit generators. These constructions do not rely on full character theory and have practical implementations in computational algebra, coding theory, and the explicit structural analysis of group rings (Gelder et al., 2013, Dai et al., 2022).

In dihedral and quaternionic group algebras, explicit primitive decompositions are realized using matrix representations and trigonometric identities derived from character table orthogonality, and the preimages of matrix units correspond bijectively to minimal ideals (Dai et al., 2022).

3. Inversion Formulas and Canonical Bases in Trivial Source Algebras

The construction of primitive idempotents in the trivial source algebra follows a Gluck–Yoshida–type inversion, expressing each primitive idempotent $e_{Q,[s]}$ as a linear combination of canonical basis elements $[N_{P,\varphi}]$: $$e_{Q,[s]} = \frac{1}{|N_G(Q)|} \sum_{P \leq Q,\, g \in N_G(Q)_{p'}} |P|\;\mu_g(P, Q) [N_{P,\varphi}],$$ where $\mu_g(P, Q)$ is a Möbius function related to reduced Euler characteristics of certain $p$-subgroup posets and irreducible Brauer characters $\varphi$ run over normalizers $N_G(P)$. This inversion provides a direct combinatorial and character-theoretic description of the minimal ideals and connects the expansion of canonical modules to both local subgroup data and topological poset invariants. Matrix representations of morphisms between rings (e.g., linearization maps) are likewise given explicitly in terms of character values and reduced Euler characteristics, establishing a tight link between combinatorics, character theory, and primitive idempotents (Barker, 2018).

4. Modular and Categorical Contexts

Modular Representation Theory

For finite semisimple algebras (e.g., Schur rings, trivial source algebras), primitive idempotents correspond to minimal two-sided ideals, directly yielding the Wedderburn decomposition: $R = \bigoplus_i R e_i,$ where $e_i$ are primitive central idempotents, usually constructed explicitly from the semilattice of normal subgroups or via character-theoretic means. In the case of Schur rings over cyclic groups, these idempotents are indexed by normal subgroups and are formulated in terms of products of normalized subgroup sums, providing a field-theoretic description of minimal components (Misseldine, 2013).

Algebraic and Differential Categories

In the category of character sheaves on solvable groups, minimal idempotents correspond to "admissible pairs," with conjectural equivalence for more general groups. These idempotents partition the triangulated monoidal category into manageable blocks, central to categorical representation theory and the realization of character sheaves (Deshpande, 2013).

5. Generalized and Topological Settings

Semigroups and Pseudo-Finite Structures

In pseudo-finite semigroups with a completely simple minimal ideal $K$, idempotents in $K$ are automatically primitive: each generates a minimal left (or right) ideal and sits in a unique $\mathcal{L}$-class (or $\mathcal{R}$-class). The Rees matrix structure allows one to index these minimal ideals and coordinate their interaction via group-like mechanisms. Conditions for pseudo-finiteness (e.g., finiteness of certain index sets) ensure that only finitely many primitive idempotents arise, controlling the combinatorics of the minimal ideal and its representation (Gould et al., 2022).

C*-Algebras and Dixmier Ideals

In general C*-algebras, an ideal is semiprime if and only if it is idempotent (i.e., $I^2 = I$). Such semiprime/primitive ideals are necessarily self-adjoint. The theory of Dixmier ideals provides a systematic lens for squeezing arbitrary ideals between "regular" (hereditary, strongly invariant) ones, and the closure under positive square roots is a key feature of these structures. The Pedersen ideal exemplifies a norm-dense, semiprime ideal populated by minimal idempotent-like elements analogous to primitive idempotents in the algebraic (non-topological) sense (Gardella et al., 2023, Kaftal et al., 2017).

Central Idempotents in Compactifications

In topological algebra, primitive idempotents in compactifications such as the weakly almost periodic (WAP) compactification $G^W$ correspond to unique elements in minimal ideals associated to minimal group compactifications. These central idempotents act as projections onto primary components of representation spaces and function algebras, canonically splitting WAP algebras and Fourier–Stieltjes algebras according to invariant topologies (Spronk, 2018).

6. Polynomial and Module-Theoretic Frameworks

Two-Dimensional Algebras

For two-dimensional algebras, primitive idempotents are parameterized by the solutions to specific cubic or quadratic polynomials determined by the structure constants. Each nontrivial minimal left ideal arises as the span of a primitive idempotent, and the classification of such algebras rests upon the nature and multiplicity of solutions to these defining polynomials (Ahmed et al., 2018).

Quotient Rings and Coding Theory

In the context of rings $R[X]/\langle g \rangle$, for a coprime factorization $g = \prod_{i=1}^r g_i$, primitive idempotents are constructed by Bézout identities: $e_i = v_i(x)\,\hat{g}_i(x),$ with $u_i(x)g_i(x) + v_i(x)\hat{g}_i(x) = 1$, $\hat{g}_i = g/g_i$. These idempotents decompose the ring as a direct sum of indecomposable local rings, and any module/ideal (such as a constacyclic code) is coordinatized via these minimal components (Charkani et al., 2019).

7. Generalized Notions: Quasi-Absorbing and A-Primitive Elements in Semigroups

In semigroup theory, the primitive idempotents relative to a given ideal $A$ are those that are minimal above $A$ in the Rees order: an idempotent $e \notin A$ is $A$-primitive if every idempotent $f\leq_\mathcal{H}e$ is either $e$ itself or lies in $A$. This perspective generalizes the classical notion to incorporate the "pruning" effect of absorbing/quasi-absorbing elements, which is operationally necessary in contexts such as set optimization, where minimal "non-trivial" idempotents describe efficient frontiers or structural limits (Hager et al., 2023).

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In summary, primitive idempotents of minimal ideals serve as irreducible, often canonically constructed, generators of minimal (left/right/two-sided) ideals in rings, algebras, monoids, and semigroups. Their explicit construction, classification, and interaction with symmetry groups, structural anti-involutions, and poset invariants provide a unifying framework for deep theoretical analysis and computational applications across algebra, representation theory, coding theory, operator algebras, and categorical settings.