Conjecture Proposer Module in Algebra

• Conjecture Proposer Module is an algorithmic framework that automates the generation of mathematical conjectures by formalizing relationships among invariants, symmetries, and module properties.
• It employs data-driven and category-theoretic techniques to parameterize simple modules over functor categories, offering precise templates for module classification.
• Case studies using Green biset functors illustrate its practical applications in constructing quotient algebras, validating template constructions, and guiding computational tests.

A Conjecture Proposer Module is an algorithmic or structural framework designed to automate or systematize the generation of mathematical conjectures, typically by formalizing relationships among invariants, symmetries, or module-theoretic properties within a given mathematical context. In modern mathematics, such modules synthesize data-driven, combinatorial, and category-theoretic techniques to propose generalizations, parameterizations, or explicit bounds—often providing precise templates for the organization, refinement, and anticipated structure of new mathematical conjectures.

1. Parameterization Frameworks for Simple Modules

A prototypical function of a conjecture proposer module in abstract algebra is the systematic classification of simple modules over functor categories via explicit parameter sets. For Green biset functors $A$, Bouc’s conjecture posits that simple $A$-modules are in bijection with equivalence classes of pairs $(H,V)$, with $H$ a finite group and $V$ a simple module over the "local" algebra $\widehat{A}(H)$, defined as

$\widehat{A}(H) = A(H \times H) / I_A(H),$

where $I_A(H)$ is generated by all composites $a \circ b$ with $a \in A(H \times K)$ and $b \in A(K \times H)$ for $|K| < |H|$. The set of parameterizing pairs is

$$\mathcal{S}_A = \{(H,V) : \widehat{A}(H) \neq 0,~ V \text{ simple } \widehat{A}(H)\text{-module} \}/\sim,$$

with the equivalence $(H, V) \sim (G, W)$ if there is an isomorphism $\varphi: H \rightarrow G$ such that $V \cong {}^\varphi W$.

This paradigm generalizes classical results for Burnside and representation ring functors, recovering known parameterizations by $\mathrm{ROut}(H)$ or group cohomology in special cases. For a conjecture proposer module, the explicit construction of $I_A(H)$ and computation of the nontriviality of $\widehat{A}(H)$ for structural classes of groups (cyclic, abelian, groups with specific ramification properties) forms the backbone of parameter synthesis.

2. Constructive Examples and Validation

Constructive examples supply both experimental evidence and template-building data. Three canonical Green biset functors illustrate this approach:

• For $kR_{}$ (rational representations), one obtains nontrivial $\widehat{kR}(H)$ only for cyclic $H$, with $V$ a primitive module over $k(\mathbb{Z}/|H|\mathbb{Z})^\times$.
• The functor $R_{}$ (complex representations) yields only a trivial simple module, as the tensor structure and bilinear properties force all nonzero modules to be trivial at the minimal group (the trivial group).
• Yoneda–Dress constructions $RB_C$—where $C$ is a cyclic group of prime order—demonstrate that the category of simple modules again aligns with parametrizations by minimal groups, with all minimal groups isomorphic and module structure governed by $\widehat{RB_C}(H)$.

In each case, the “local” algebra setting confirms conjectural behavior, producing a finite list of equivalence classes. When applied in a conjecture proposer module, these families serve as templates for algorithmically recognizing other functors which (possibly) admit similar module-theoretic classification.

3. Structural Proofs and Category Theoretic Localization

Proofs of such conjectures exploit the idea that minimal groups for simple modules are unique up to isomorphism, and that the module structure for any simple $A$-module can be reconstructed from the data at this minimal group. This is formalized by functors of the form

$$L_{H,V}(G) = A(G \times H) \otimes_{A(H \times H)} V,$$

with the unique simple quotient $S_{H,V}$ constructed via the appropriate quotient by maximal submodules generated by post-composition vanishing elements.

Key to this approach are category-theoretic arguments: by localizing at minimal groups and analyzing the structure of morphisms between them, one ensures the one-to-one correspondence between simple modules and “seeds” $(H,V)$. In practice, equivalence classes under group isomorphism and module twisting become intrinsic data objects, and the calculation of the quotient algebra $\widehat{A}(H)$ follows explicit functorial workflows.

4. Generalization Beyond Classical Biset Functors

Bouc’s conjecture generalizes the classical classification for the Burnside ring, where parameterization by pairs $(H, V)$ is governed by $\mathrm{ROut}(H)$: $\widehat{RB}(H) \cong \mathrm{ROut}(H).$ The extension to arbitrary Green biset functors $A$ links the additional algebraic structure derived from monoidal (tensor product) operations to the local analysis captured by $\widehat{A}(H)$. This shift has significant implications: it unifies the classification of modules over a diverse array of functors (including rational and complex representation rings, as well as twisted versions like the Yoneda–Dress functors) via a single parametric principle.

From a conjecture proposer module perspective, this abstraction serves as a pattern for extending classification strategies to more general or exotic functors, by (a) identifying local objects where “support” is concentrated, and (b) transferring ring-theoretic invariants into module-combinatorial frameworks for prediction.

5. Open Problems, Computational Directions, and “Proposing” New Conjectures

Challenges and future directions remain central to the “proposing” aspect:

• Classification when minimal groups are not unique up to isomorphism, or when $I_A(H)$ unexpectedly vanishes.
• Behavior of quotient algebras $\widehat{A}(H)$ in more general cases (for instance, Burnside functors where $\widehat{RB}(H)$ may vanish).
• Complications in the Yoneda–Dress construction for non-prime cyclic $C$, where the factorization property fails.
• Development of computational (e.g., GAP-based) tools to suggest or test candidate pairs $(H, V)$ given the explicit data of a Green biset functor $A$.
• Transfer of structural insights to broader areas such as Mackey functor theory, modular representation theory, and the paper of saturated fusion systems.

For automated conjecture proposer systems, these tasks motivate the algorithmic analysis of quotient ideals, the search for minimal representatives in the module category, and meta-level proposal of new conjectural relationships grounded in explicit calculations and category-theoretic properties.

6. Schematic Formulas and Algorithmic Ingredients

Universal aspects of a conjecture proposer module for this context include:

• Definition and computation:

$$I_A(H) = \langle a \circ b: a \in A(H \times K),~ b \in A(K \times H),~ |K| < |H| \rangle,$$

$\widehat{A}(H) = A(H \times H)/I_A(H).$

• Parametrization set (modulo equivalence by group isomorphism):

$$\mathcal{S}_A = \{(H, V) : \widehat{A}(H) \neq 0,~ V \text{ simple } \widehat{A}(H)\text{-mod}\}/\sim.$$

• Functorial construction for new simple modules:

$$L_{H,V}(G) = A(G \times H) \otimes_{A(H \times H)} V,$$

unique simple quotient $S_{H,V}$ by annihilation of “post-composition” maximal submodules.

These components distill the inductive, algebraic, and classification-based logic that underpins successful deployment of automated conjecture proposing in module categories.

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The Conjecture Proposer Module concept thus encapsulates both a meta-methodology and practical infrastructure for classifying and predicting the structure of simple modules over highly nontrivial functor categories. Its grounding in parametrization by local algebras, explicit functorial construction, and category-theoretic localization is directly evident in the Green biset functor framework and serves as a blueprint for generalized module-theoretic classification and automatable conjecture generation (Romero, 2012).