| 1. Domain | 1.1 Scope of the Domain | Boundaries | The range of phenomena the science includes and excludes. | Studies how algebraic structures—groups, algebras, Lie algebras—act linearly on vector spaces. Includes group representations, modules over algebras, characters, irreducible decompositions, tensor products, induced representations, and geometric representation frameworks. Excludes nonlinear actions and structure theories unrelated to linearization. |
| | Scale | The spatial, temporal, or organizational level at which the science operates (e.g., quantum, cellular, social, cosmic). | Operates across algebraic, geometric, and analytic scales: finite-dimensional vs infinite-dimensional representations, local/global symmetries, decomposition into irreducibles, spectral-type behaviors, and topological/analytic structures in continuous groups. |
| 1.2 Ontological Commitments | Entities | The kinds of things assumed to exist within the domain (particles, organisms, agents, fields, etc.). | Vectors, linear maps, representation homomorphisms, group/algebra elements, invariant subspaces, irreducible components, characters, weights, roots, modules, basis elements, intertwiners, tensor factors. |
| | Properties | The fundamental attributes these entities possess (mass, charge, genotype, preference, etc.). | Linearity of action; invariance of subspaces; reducibility/irreducibility; complete reducibility (in semisimple settings); character orthogonality relations; weight-space decomposition; Schur’s Lemma properties; highest/lowest weights; multiplicities. |
| | Categories | The basic ontological types used to classify domain elements (substances, processes, relations, structures). | Representations of finite groups; Lie group/Lie algebra representations; modules over associative algebras; unitary representations; irreducible representations; reducible but indecomposable representations; semisimple categories; monoidal categories; tensor representations. |
| 1.3 State-Variables | Variables | The measurable or definable properties that describe system conditions. | Choice of basis; matrix form of representation; character values; dimension; irreducible decomposition components; weight vectors; highest-weight parameters; intertwiner maps; tensor-product multiplicities. |
| | Parameterization | How variables encode and represent the system’s state. | Encoded via matrices, modules, characters, highest-weight diagrams, weight lattices, root systems, decomposition tables, representation rings, tensor-decomposition coefficients (Clebsch–Gordan, Littlewood–Richardson). |
| 1.4 Admissible Idealizations | Simplifications | Conceptual reductions used to make the domain tractable (point masses, rational agents, perfect gases). | Focusing on semisimple categories; assuming complete reducibility; restricting to finite-dimensional representations; using orthonormal bases; ignoring pathological indecomposable structures; treating compact groups as fully unitary. |
| | Validity Conditions | The limits and contexts in which idealizations hold or break down. | Fail in non-semisimple algebras; infinite-dimensional representations with analytic subtleties; non-compact groups lacking complete reducibility; modular representation theory where usual character theory breaks down; wild representation types. |
| 1.5 Domain Assumptions | Structural Assumptions | Background ontological stances such as determinism, continuity, randomness, discreteness. | Actions are linear; homomorphisms preserve structure; decomposition reflects symmetry; characters and weights encode structural information; tensor products behave functorially; representations classify symmetry behavior. |
| | Implicit Commitments | Unstated but necessary assumptions that shape the field’s conceptual structure. | Assumes existence of bases compatible with structure; assumes decomposition methods apply; assumes character theory is meaningful (in semisimple cases); assumes tensor categories behave associatively and functorially; assumes duals exist. |
| 1.6 Internal Coherence Requirements | Consistency | The demand that domain concepts do not contradict one another. | Representation homomorphisms must respect algebraic operations; decompositions must be consistent with invariant subspaces; character relations must hold; tensor products must preserve representation axioms; intertwiners must satisfy categorical coherence. |
| | Compatibility | The requirement that entities, variables, and assumptions fit together into a unified descriptive framework. | Requires harmony among modules, tensor products, characters, decomposition rules, weight structures, functorial relationships, and symmetry phenomena across algebra, geometry, and analysis. |
| 2. Evidence Layer | 2.1 Observable Phenomena | Observables | The aspects of the domain that can produce detectable signals accessible to measurement. | Matrix behavior under group/algebra actions; invariance of subspaces; decomposition into irreducibles; eigenvalues/eigenvectors of representing matrices; character values; multiplicity patterns; tensor-product decomposition outcomes; weight-space structure; branching rules. |
| | Detection Limits | The boundaries of what can be resolved or sensed by current instruments or methods. | Limited by inability to decompose representations of wild algebras; difficulty computing irreducibles for large groups; non-unitary representations obscuring spectral clarity; analytic obstructions in infinite-dimensional cases; insufficient resolution for continuous spectrum. |
| 2.2 Measurement Systems | Units | Standardized quantifications (meters, seconds, volts, decibels, dollars, etc.) necessary for consistent comparison. | Dimension of representation; character values; multiplicities of irreducible components; rank of weight spaces; highest-weight parameters; matrix entries; norms in unitary representations; Casimir eigenvalues. |
| | Instruments | Devices and tools (microscopes, spectrometers, sensors, surveys, detectors) used to produce measurements. | Character tables; matrix representation calculators; Lie algebra root/weight computation tools; tensor decomposition algorithms (e.g., Littlewood–Richardson); numerical diagonalizers; spectral analyzers; computational algebra systems (GAP, Magma, Sage). |
| 2.3 Operational Definitions | Definitions | Terms defined by specific measurement procedures, ensuring empirical clarity. | Representation defined as homomorphism to GL(V); irreducibility defined via lack of invariant subspaces; character defined as trace of representing matrices; equivalence defined by conjugation; weights defined via eigenvalues of Cartan subalgebra elements; intertwiners defined as structure-preserving maps. |
| | Procedures | The explicit steps required to perform a measurement in a reproducible way. | Constructing matrix models from generators; computing characters; decomposing representations using orthogonality relations or numerical diagonalization; computing weight diagrams; determining highest weights; evaluating tensor products; finding intertwiners; restricting representations to subgroups/subalgebras. |
| 2.4 Data Acquisition | Protocols | Formal processes for gathering data under controlled or standardized conditions. | Standardized character-table lookup; canonical matrix construction for groups/algebras; systematic generation of weight diagrams; consistent tensor-product computations; analytic continuation procedures in infinite-dimensional cases; structured use of branching rules. |
| | Sampling | Rules determining which subset of the domain is measured and how representative it is. | Sampling representations of small groups; sampling over irreducible families; selecting random weight vectors; sampling tensor products; selecting different bases for matrix representations; sampling decompositions across varying highest weights. |
| 2.5 Data Character & Format | Data Types | The form raw evidence takes (time series, spectra, images, counts, qualitative records). | Matrices; character tables; decomposition lists; weight lattices; branching diagrams; tensor-product multiplicity tables; eigenvalue/eigenvector sets; Casimir eigenvalue lists; intertwiner spaces. |
| | Resolution | The granularity or precision with which data is captured. | Determined by matrix size; precision of eigenvalue computations; granularity of weight-lattice sampling; completeness of character tables; accuracy of numerical diagonalization; resolution limits in infinite-dimensional spectra. |
| 2.6 Reliability & Calibration | Calibration | Adjustment procedures ensuring instruments produce accurate results. | Verifying characters with orthogonality relations; cross-checking decompositions using independent methods; validating Casimir eigenvalues; confirming intertwiner properties; checking consistency across bases; ensuring tensor decompositions satisfy associativity constraints. |
| | Error Characterization | Identification and quantification of noise, uncertainty, bias, and measurement error. | Incorrect decomposition; numerical instability in eigenvalues; misclassified highest weights; faulty character computations; incorrect branching rules; errors in tensor-product multiplicities; basis-dependent representational mistakes; failure in detecting invariant subspaces. |
| 3. Structural Layer | 3.1 Patterns & Regularities | Laws / Relations | Stable, repeatable patterns governing how observables behave across conditions. | Linearization of group/algebra actions; decomposition into irreducibles (when semisimple); Schur orthogonality relations; tensor-product decomposition laws (Clebsch–Gordan, Littlewood–Richardson); weight-space decomposition patterns; induced/restricted representation relations; reciprocity laws (Frobenius reciprocity). |
| | Invariants | Quantities or properties that remain constant under transformations (symmetries, conservation laws). | Character values; dimensions; multiplicities of irreducibles; weights; highest weights; Casimir eigenvalues; central characters; equivalence class of representation; categorical invariants in semisimple tensor categories. |
| 3.2 Causal Architecture | Mechanisms | Underlying processes or structures that produce the observed regularities. | Group/algebra actions producing invariant subspaces; homomorphisms determining module structure; weight decomposition driven by Cartan subalgebra; tensor products generating new representations; induction and restriction mechanisms transforming representations along subgroup chains; functorial constructions generating structure-preserving maps. |
| | Pathways | Organized sequences of interactions forming a causal chain or network. | Representation → compute characters → decompose into irreducibles; module → find invariant subspaces → classify irreducibility; group → subgroup restriction → branching rules; tensor product → decomposition → compute multiplicities; Lie algebra → root system → weight diagram → highest-weight classification. |
| 3.3 Theoretical Vocabulary | Concepts | Core terms that encode the domain’s structure (force, gene, equilibrium, field). | Representation, module, irreducible representation, character, intertwiner, invariant subspace, highest weight, weight lattice, root system, tensor product, induction, restriction, decomposition, Casimir operator, Schur’s Lemma, semisimplicity. |
| | Classifications | Taxonomies, categories, or typologies that organize entities and relations. | Finite vs infinite-dimensional representations; unitary vs non-unitary; reducible vs irreducible; semisimple vs non-semisimple; representations of finite groups, Lie groups, Lie algebras, associative algebras; highest-weight modules; projective representations; modular representations (positive characteristic). |
| 3.4 Formal Representations | Equations | Mathematical constructs expressing laws, relations, or mechanisms. | Homomorphism relation: ρ(g₁g₂)=ρ(g₁)ρ(g₂); character equation: χ(g)=trace(ρ(g)); orthogonality relations: ⟨χᵢ,χⱼ⟩=δᵢⱼ (for semisimple categories); weight-space equations: H·v=λ(H)v; tensor decomposition equations; Casimir eigenvalue equations; highest-weight defining relations. |
| | Models | Structured representations—mathematical, computational, or conceptual—used to predict and explain phenomena. | Matrix representations; module diagrams; weight diagrams and root lattices; character tables; Bratteli diagrams; category-theoretic models (monoidal categories, functors); geometric models via orbits and equivariant sheaves; representation rings. |
| 3.5 Idealized Structures | Simplified Models | Purposeful abstractions that capture essential dynamics while omitting irrelevant detail. | Completely reducible settings (finite groups over ℂ, compact Lie groups); diagonalizable operators; orthonormal bases; finite tensor categories; simple highest-weight modules; ignoring non-semisimple or wild-type behavior; restricting to low-dimensional modules. |
| | Limit Conditions | Regimes where specific models or approximations hold (classical vs. quantum, linear vs. nonlinear). | Simplifications fail in modular representation theory; infinite-dimensional representations require analytic control; non-semisimple algebras defy decomposition; wild representation types prevent classification; representations of non-compact groups require harmonic analysis; tensor decompositions may be infinitely large. |
| 3.6 Integrative Frameworks | Unifying Theories | Higher-order structures that connect disparate laws or mechanisms under a coherent whole. | Tannakian duality linking tensor categories and group reconstruction; highest-weight theory unifying Lie algebra/classical group representations; character theory unifying finite-group representation classification; geometric representation theory linking algebraic geometry and modules; Langlands correspondence in advanced settings. |
| | Interdisciplinary Links | Points where the theory connects to adjacent sciences or larger explanatory systems. | Physics (quantum mechanics, particle symmetries, gauge theory); chemistry (molecular symmetries); number theory (automorphic forms, Langlands program); algebraic geometry (perverse sheaves, geometric Satake); computer science (group algorithms, coding theory); topology (equivariant cohomology). |
| 4. Method Layer | 4.1 Inquiry Design | Experimental Design | Structured plans for manipulating variables to test causal claims. | Varying bases for representations; modifying generating sets for groups/algebras; constructing alternative matrix representations; altering weight choices or Cartan subalgebras; changing tensor-product inputs; modifying subgroup chains in restriction/induction; introducing different highest-weight parameters. |
| | Observational Design | Systematic approaches for gathering non-manipulated data (surveys, field studies, natural experiments). | Observing natural invariant-subspace emergence; monitoring decomposition of representations under fixed symmetry groups; observing branching behavior under subgroup restriction; tracking weight shifts under Lie algebra actions; watching multiplicities appear in tensor products without altering structural definitions. |
| 4.2 Testing & Validation | Hypothesis Testing | Procedures for evaluating whether evidence supports or contradicts specific claims. | Testing irreducibility via invariants; checking Schur orthogonality; validating character identities; verifying highest-weight predictions; testing decomposition via tensor-product rules; confirming equivalences via intertwiner existence; validating functoriality in categorical settings. |
| | Replication | The requirement that results be independently reproducible under similar conditions. | Recomputing characters; repeating decompositions using multiple algorithms; recalculating weight diagrams; recomputing tensor-product multiplicities; verifying Gelfand–Tsetlin patterns; confirming branching rules with alternative subgroup chains; repeating Casimir eigenvalue computations. |
| 4.3 Inference & Evaluation | Statistical Inference | Rules for drawing conclusions from noisy or incomplete data. | Analyzing distribution of irreducible multiplicities; assessing frequency of particular weights in random representations; evaluating stability of eigenstructures under perturbations; comparing tensor-product multiplicity growth; studying character-value distributions across group elements. |
| | Model Comparison | Criteria (fit, simplicity, predictive accuracy, robustness) used to evaluate competing models. | Comparing matrix vs. abstract module models; comparing representations of different but related groups; contrasting Lie algebra vs. Lie group representations; comparing decompositions under different bases; evaluating different tensor categories; comparing unitary vs. nonunitary models. |
| 4.4 Error Management | Error Analysis | Identification and quantification of random and systematic errors. | Detecting incorrect decomposition; identifying wrong character computations; misclassification of highest weights; errors in branching rules; incorrect intertwiner constructions; numerical instability in eigenvalue-based decompositions; mistakes in weight-space identification. |
| | Bias Control | Methods for minimizing subjective, instrumental, or procedural biases. | Avoiding basis-dependent artifacts; preventing bias toward semisimple cases; ensuring sampling across reducible and indecomposable modules; avoiding selective decomposition of “nice” representations; controlling for group size/complexity; avoiding bias in tensor-product selection. |
| 4.5 Adjudication & Revision | Peer Scrutiny | Collective evaluation of claims through critique, review, and debate. | Reviewing decomposition proofs; auditing character-table derivations; checking correctness of highest-weight arguments; cross-validating branching rules; reviewing tensor-category coherence; rechecking spectral predictions; comparing independent computational pipelines. |
| | Theory Revision | Procedures for modifying, replacing, or discarding models based on new evidence. | Updating decomposition rules; correcting character identities; refining highest-weight classifications; modifying induction/restriction frameworks; incorporating new categorical results; adjusting tensor-product rules; revising assumptions in modular/positive-characteristic settings. |
| 4.6 Integrity Conditions | Transparency | Requirements to disclose methods, data, assumptions, and limitations. | Full disclosure of bases, generating sets, matrix forms, computation methods, decomposition algorithms, tensor conventions, and normalization choices; explicit acknowledgement of semisimple vs. non-semisimple assumptions. |
| | Ethical Standards | Norms ensuring responsible conduct in experimentation, data handling, and publication. | Honest reporting of ambiguities or non-uniqueness in decompositions; ensuring reproducibility; disclosing limitations in computational methods; avoiding overclaims in non-semisimple contexts; maintaining rigor in categorical and Lie-theoretic statements. |