feat: add the representation theory roadmap family

[kim-em]

This PR adds a family of eleven representation-theory roadmaps under TauCetiRoadmap/RepresentationTheory/ The family covers:

• semisimple algebras and Artin-Wedderburn structure theory
• the character theory of finite groups, up to the Burnside-Dixon-Schneider algorithm that computes character tables over an exact cyclotomic type
• induction, restriction, and Mackey theory, with Artin and Brauer induction, projective representations, and the Schur multiplier
• root systems, Weyl groups, and the Cartan-Killing classification
• the highest-weight theory of semisimple Lie algebras through the Weyl character and dimension formulas, the Harish-Chandra isomorphism, and the explicit construction of the exceptional algebras
• the finite-dimensional representations of the classical groups, with the Weyl construction, Schur polynomials, and Gelfand-Tsetlin bases, the symmetric group, Specht modules, and Schur-Weyl duality for the general linear, orthogonal, and symplectic groups
• compact groups, Haar measure, and the Peter-Weyl theorem
• Lie groups and the Lie algebra correspondence
• Clifford algebras, the Pin and Spin groups, and the spin representations that supply the fundamental representations of types B and D
• quiver representations, Gabriel's theorem, and Auslander-Reiten theory.

The roadmaps cite the neighbouring reductive-groups, orthogonal-L2-bases, pivotal-spherical, and Temperley-Lieb roadmaps where relevant.

🤖 Prepared with Claude Code

[mrdouglasny]

This is a supervised, question-driven review; the blocking items are below, full analysis available in https://github.com/mrdouglasny/TauCeti-reviews-pr62.

🤖 Models Claude Opus 4.8 and Codex (GPT-5.6) suggest:

**Overall the family is strong and largely formalization-ready: the mathematics, layering, references (cited per-layer in every roadmap), and Mathlib-overlap accounting are consistently accurate. The systematic risk is signature faithfulness — Suggested.lean targets that would build green while being false or vacuous. We found 15 such blocking defects across 9 of the 11 roadmaps (only SchurWeyl and RootSystems are blocking-free), plus ~26 should-fix items and 4 coverage requirements. Every blocking item below was adversarially verified (Codex, with counterexamples).

🔴 Blocking (false or vacuous as written)

• SpinRepresentations
  • card_ker_spinToSpecialOrthogonal — false in dimension 0 (V=0 ⇒ Pin/Spin and SO(0) trivial ⇒ kernel is 1, not 2).
  • soEquivBivector — false in characteristic 2 (R=𝔽₂, n=1: ⋀²R=0 but so(1,𝔽₂)≅𝔽₂); needs ≃ₗ⁅R⁆ + [Invertible 2].
  • spinAction/spinRep — target can be empty (W=0: no unital Mat₂(ℂ)→ℂ); needs a maximal-isotropic polarization.
• CharacterTheory
  • irreducibleCharacters_span — false: irreducible characters span ClassFunction, not all of G→k (⊤ ≤ span fails for nonabelian G).
  • classMultMatrix vs centralCharacterTable_eigenvector — transpose mismatch (a(i,k,j) vs a(i,j,k); S₃ witness a_{T,T,C}=3 ≠ a_{T,C,T}=2).
  • characterPairing_nondegenerate — false if char ∣ |G| (pairing ≡ 0); needs [Invertible (card G : k)].
• QuiverRepresentations
  • dimVector_reflectionFunctor — vacuous: IsEmpty (Sᵢ ⟶ M) is unsatisfiable in a preadditive category.
  • exists_projectiveCover — false over arbitrary [Ring A] (needs semiperfect/Artinian).
  • exists_almostSplitSequence — false on the whole functor category (AR existence is for finitely-presented reps; Paquette 2011).
• ClassicalGroups — schurFunctor_iso_irreducible, character_irreducible_eq_schurPoly, and gtPatternEquivSSYT all false without the row bound ℓ(μ) ≤ n.
• InductionRestriction — brauer_characterization false without a class-function hypothesis, even over ℂ (S₃ counterexample); the splitting-field fix alone does not repair it.
• SemisimpleAlgebras — card_blocks_eq false: zero-size Wedderburn blocks break the count (needs ∀ i, NeZero (d i)).
• LieHighestWeight — sl2_finrank_of_hasPrimitiveVector / sl2_irreducible_ext false without "the triple spans L" (sl₃-adjoint counterexample).
• CompactGroups — schur_orthogonality_self false (conjugation on the wrong factor under Mathlib's inner-product convention).
• LieGroups — weyl_integration_formula false: μG, μT, weylDensity are free unconstrained parameters.

🟠 Should-fix and coverage (detail in the bundle)

~26 should-fix items (missing hypotheses, vacuous Prop := sorry anti-vacuity predicates, too-weak isos), plus four coverage requirements written up as standalone notes: (1) pin the representation ring R(G) + injective character ring-hom (tensor-product decomposition currently has no spec — only the additive shadow is pinned); (2) pin the executable character-table summit's cyclotomic bridge (Cyclotomic e / embedding / lift); (3) extend the Frobenius–Schur reality trichotomy to compact groups (present for finite groups only — needed for spinor/matter reality types); (4) add a "Scope & boundaries" paragraph to the family README (the deliberate exclusions — modular theory, noncompact rep theory, affine types — are nowhere stated).

Recurring themes: faithfulness-over-falsity (a green build won't catch these); unpinned cross-roadmap interfaces promised in prose but not machine-checked; and Prop := sorry predicates that leave a summit only as strong as an unwritten definition.

📦 Full analysis (linked)

Supporting material — too long for this comment — is at https://github.com/mrdouglasny/TauCeti-reviews-pr62:

• reviews/ — per-roadmap findings (blocking + should-fix + notes) and the family synthesis; 01_INDEPENDENT_ADVERSARIAL_AUDIT.review.md is Codex's independent audit.
• VERIFICATION.md — Codex's counterexample-level verification of the original blockers.
• OVERVIEW.md — structures defined, Lean design choices, one major theorem per roadmap, and a per-definition universal-property / canonicity scorecard.
• DEFINITIONS.md — a checklist of all 238 defs, each linking to its exact source line.
• APPLICATIONS.md / APPLICATIONS_AG.md — a downstream-support audit (mathematical physics; algebraic geometry, Fulton–Harris) of whether the foundations expose what good downstream definitions will need.
• comments/ — the four coverage requirements above, as postable notes.
• roadmaps/ — the reviewed PR #62 snapshot (head 37f34c8), so each review sits next to the roadmap it reviews.

Happy to discuss or open per-finding threads.