Fulton, Harris — *Representation Theory: A First Course* (Fast Track 3.11) — Audit + Gap Plan

Book: William Fulton, Joe Harris, Representation Theory: A First Course (Graduate Texts in Mathematics 129 / Readings in Mathematics, Springer 1991, xv + 551 pp., ISBN 0-387-97527-6). Standard reference; commercial title (BUY in docs/catalogs/FASTTRACK_BOOKLIST.md row 3.11).

Fast Track entry: 3.11. The canonical graduate representation-theory text of the modern era — distinguished from the Fast Track's other rep-theory slots (2.02 Woit QM, Groups and Representations; 3.12/3.13/3.15 Serre's three slim volumes; 3.16 Diaconis) by being comprehensive and extremely concrete: finite groups (with the $S_n$ examples worked out in painstaking detail) → compact Lie groups → classical complex Lie algebras (with Dynkin diagrams introduced as classification tools, not abstract decoration) → general semisimple theory. The Fast Track source page describes it as the "complex case" classification text (Serre's Complex Semisimple Lie Algebras being the ludicrously brief companion).

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of deepenings + a small set of new units so that Representation Theory: A First Course (FH hereafter) is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

REDUCED audit. No local PDF in reference/textbooks-extra/ (only the cover image at reference/fast-track/images/Fulton-Harris-Rep-Theory-...jpg). Springer preview is gated; Google Books preview is gated. This audit works from (a) the public TOC structure of GTM 129 (well-documented and referenced by the Fast Track source page §3.11), (b) the Codex's existing 26 shipped units in content/07-representation-theory/, and (c) the originator literature. A full line-number audit is deferred until a PDF is acquired. This is consistent with the audit-stub convention used for Brown-Higgins-Sivera (1.05a).

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§1 What FH is for

FH is the canonical concrete graduate rep-theory text. Where Serre's Linear Representations of Finite Groups (FT 3.15) is the slim crystalline finite-group treatment, where Knapp's Lie Groups Beyond an Introduction gives the structure-theoretic real-form sweep, and where Hall's Lie Groups, Lie Algebras, and Representations keeps everything matrix-explicit at upper-undergraduate level — FH is the bridge: rigorous like Serre, matrix-explicit like Hall, but covering the entire arc from $S_n$ to the Weyl character formula in one volume with every classical Lie algebra (${\mathfrak{sl}}_n,{\mathfrak{sp}}_{2n},{\mathfrak{so}}_n$) worked out by hand before the general theory is invoked. This worked-example-first pedagogy is the book's signature contribution.

Distinctive contributions, in roughly the order FH develops them:

1. Character theory of finite groups, with $S_n$ as the running case study. FH §§1–5: definitions, Schur's lemma, character orthogonality, induced representations, Frobenius reciprocity. §4 introduces Young diagrams and Young symmetrisers to construct the Specht modules of $S_n$ explicitly — and §§5 the Frobenius character formula, hook length formula, and Schur-Weyl duality. The Frobenius character formula [Frobenius 1900] is given a fully self-contained proof using only the orthogonality relations. Originator citation: Schur 1901 PhD thesis (Schur-Weyl duality); Frobenius 1896/1900 (the character theory programme he initiated by correspondence with Dedekind).

2. Lie groups and Lie algebras introduced through the matrix examples first. FH §§7–9: ${\mathfrak{sl}}_2\mathbb{C}$ is worked out completely — its irreducible representations $V_n$ classified by highest weight $n$, the Casimir element constructed explicitly, the character ${\chi }_n(t)=(t^{n+1}-t^{-n-1})/(t-t^{-1})$ derived by weight counting. This is the only place in the literature where the reader is invited to rediscover the abstract theory from a single worked example.

3. The ${\mathfrak{sl}}_3\mathbb{C}$ chapter (FH §§11–13) — the pedagogical centrepiece. Roots, weights, the Weyl group $S_3$ acting on the weight lattice, the fundamental chamber, and the hexagonal-honeycomb pictures that have appeared in every subsequent rep-theory exposition. By the end of §13 the reader has the entire highest-weight programme in their hands as a picture, before any general Cartan-Killing theory is invoked. This visualisation-first approach is FH's distinctive editorial choice.

4. The classical Lie algebras worked out one at a time, in order: ${\mathfrak{sl}}_n$ (§15), ${\mathfrak{sp}}_{2n}$ (§16), ${\mathfrak{so}}_{2n+1}$ and ${\mathfrak{so}}_{2n}$ (§§18–19). Each chapter constructs the root system from scratch, identifies the fundamental weights, and writes down the dimensions of the standard representations using the Weyl dimension formula before invoking it abstractly. The reader sees $A_n,B_n,C_n,D_n$ emerging as Dynkin diagrams inductively from their root systems rather than postulated.

5. Spin representations (FH §20) and the ${\mathfrak{so}}_n$ / $\mathfrak{spin}(n)$ distinction via Clifford algebras. Pin and Spin groups constructed explicitly; the half-spin representations of ${\mathfrak{so}}_{2n}$ written down in terms of Clifford generators. FH presents this as a representation-theoretic phenomenon (the spinor rep is not a rep of $\mathrm{SO}$ but is of its double cover), rather than the geometric setup of Lawson-Michelsohn (FT 1.08).

6. The general theory of semisimple Lie algebras (FH §§21–24) is finally given after the entire $A$/$B$/$C$/$D$ case has been worked out: Cartan subalgebras, root-space decomposition (Cartan's theorem, 1894), the Killing form, semisimplicity criterion, abstract Dynkin diagrams, and the classification of simple complex Lie algebras into four infinite families + five exceptionals. This example-then-theory ordering is the inverse of Humphreys' standard treatment.

7. The Weyl character formula and the Weyl integration formula (FH §§24–26). Statement, proof via the Casimir, and applications: Weyl dimension formula, Steinberg formula for multiplicities, Kostant multiplicity formula. The compact-Lie-group derivation (Weyl 1925/1926) is sketched in §26 alongside the algebraic derivation. Originator citation: Hermann Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen, I–IV," Math. Zeitschrift 23–24 (1925–26) — the foundational paper of compact-Lie-group representation theory.

8. Appendices A–F: multilinear algebra (tensor and exterior powers, treated explicitly); the symmetric algebra and symmetric functions; polynomial and rational representations of ${\mathrm{GL}}_n$ via the Schur functor; Lie's theorem, Engel's theorem, semisimple and nilpotent Jordan decompositions. These appendices are load-bearing: §A on multilinear algebra is repeatedly referenced; §B/§C on symmetric functions are the technical backbone of Schur-Weyl duality and the Littlewood-Richardson rule (which FH states but proves only in special cases — the full LR rule is referred to Macdonald).

FH is not the place to learn Lie group theory in the sense of smooth manifolds with group structure (Hall is better for that, FT 3.10) and is not the place for the deep structure theory of real forms (Knapp's Beyond an Introduction, ibid.). FH is also explicit about pushing infinite-dimensional reps (Harish-Chandra modules, unitary reps of non-compact groups) entirely outside scope. The canonical follow-ups are: Knapp for real forms; Humphreys Introduction to Lie Algebras and Representation Theory for the algebraic axiomatic treatment; Bump Lie Groups for the geometric synthesis.

Cited peer sources (≥3, per stub spec):

• J.-P. Serre, Linear Representations of Finite Groups (Springer GTM 42, 1977) — FT 3.15, the slim crystalline finite-group treatment. FH §§1–5 is a "fattened" Serre with worked $S_n$ machinery.
• B. C. Hall, Lie Groups, Lie Algebras, and Representations: An Elementary Introduction (Springer GTM 222, 2nd ed. 2015) — FT 3.10 -adjacent (already in reference/quantum-well/md/Literature/). Covers the same Lie-side material at a gentler pace, matrix-only, without finite groups. Hall is FH's most direct competitor.
• A. W. Knapp, Lie Groups Beyond an Introduction (Birkhäuser 2nd ed. 2002) — cited as tier anchor in 07.01.01-group-representation.md master tier. The structure-theoretic real-form sequel to FH.
• B. E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions (Springer GTM 203, 2nd ed.
  2001. — the canonical "FH chapter 4 expanded to a book" treatment of $S_n$, RSK, and the Frobenius character formula.

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§2 Coverage table (Codex vs FH)

Cross-referenced against the current 26 shipped units of content/07-representation-theory/ plus prerequisites pulled in from 03-modern-geometry/09-spin-geometry/. ✓ = covered, △ = partial / different framing / coverage on inspection of the unit body but not the explicit FH worked example, ✗ = not covered.

FH topic	Codex unit(s)	Status	Note
Group representation, equivalence, sub/quotient	07.01.01 group-representation	✓	Master tier cites FH and Serre as anchors.
Schur's lemma	07.01.02 schur-lemma	✓	Direct correspondent.
Character of a representation	07.01.03 character	✓
Character orthogonality relations	07.01.04 character-orthogonality	✓	First and second orthogonality.
Regular representation	07.01.05 regular-representation	✓
Tensor product of representations	07.01.06 tensor-product-of-representations	✓
Induced representation	07.01.07 induced-representation	✓
Frobenius reciprocity	07.01.08 frobenius-reciprocity	✓
Maschke's theorem (complete reducibility, char 0)	07.02.01 maschke-theorem	✓
Restriction; tensor decomposition rules	—	△	Touched in 07.01.06 but no standalone unit for Mackey's restriction / induction formulae. Gap (low).
Character table of $S_3$, $S_4$, $S_5$, $A_4$	—	✗	Gap (medium). FH §2 worked-example tables; Codex has the machinery but no worked tables. Pack-pass exercise candidate.
Symmetric group representation theory	07.05.01 symmetric-group-representation	✓
Young diagrams and tableaux	07.05.02 young-diagram	✓	Hook length formula present.
Specht modules / Young symmetrisers	07.05.03 specht-module	✓
Frobenius character formula for $S_n$	△ (referenced in 07.05.02)	△	Gap (medium-low). Mentioned but no dedicated unit; FH §4 gives a complete proof. Candidate: 07.05.04 frobenius-character-formula.
Schur-Weyl duality ${\mathrm{GL}}_d\times S_n$ on $V^{\otimes n}$	△ (stated in 07.05.01, 07.05.02)	△	Gap (high). Codex states it; FH §6 gives the full proof. Candidate: 07.05.04 schur-weyl-duality (or fold into the 07.05.04 slot).
Schur functors, polynomial reps of ${\mathrm{GL}}_n$	—	✗	Gap (medium). FH §6 + Appendix C. No Codex unit. Candidate: 07.05.05 schur-functor.
Littlewood-Richardson rule	—	✗	Gap (medium-low). FH §A.1 states it; full proof referenced to Macdonald. Candidate: short pointer unit 07.05.06 littlewood-richardson.
RSK correspondence	—	✗	Gap (low). Not in FH proper (referred to Sagan/Macdonald); Codex skip is consistent with FH but a master-tier deepening would close the Sagan-coverage hole.
Lie algebra representation	07.06.01 lie-algebra-representation	✓
Universal enveloping algebra; PBW	07.06.02 universal-enveloping-algebra	✓
Casimir element	— (touched in 07.06.07)	△	Gap (medium). FH §6 uses the Casimir constantly; Codex references it in passing. Candidate: short unit 07.06.10 casimir-element.
${\mathfrak{sl}}_2\mathbb{C}$ rep theory worked out	△ (in 07.03.01 and 07.06.06)	△	Gap (high — pedagogical centrepiece of FH). Codex has highest-weight theory and Verma modules abstractly, but no dedicated ${\mathfrak{sl}}_2$ unit walking through the $V_n$ classification, ladder operators, and the character ${\chi }_n(t)=(t^{n+1}-t^{-n-1})/(t-t^{-1})$. Candidate: 07.06.11 sl2-representations.
${\mathfrak{sl}}_3\mathbb{C}$ rep theory worked out (the hexagonal-weight pictures)	—	✗	Gap (very high — the FH centrepiece). No Codex unit. Without this, a Codex reader cannot reproduce the FH §§11–13 visual programme. Candidate: 07.06.12 sl3-representations.
Root system	07.06.03 root-system	✓
Weyl group	07.06.04 weyl-group	✓
Dynkin diagram and the $A$/$B$/$C$/$D$ classification	07.06.05 dynkin-diagram	✓
Cartan-Weyl classification of simple complex Lie algebras	07.04.01 cartan-weyl-classification	✓
Highest-weight representations	07.03.01 highest-weight-representation	✓
Verma module	07.06.06 verma-module	✓
Weyl character formula	07.06.07 weyl-character-formula	✓
Weyl dimension formula	07.06.08 weyl-dimension-formula	✓
Borel-Weil theorem (geometric realisation)	07.06.09 borel-weil-theorem	✓	Goes beyond FH (FH does not cover the geometric realisation; this is FH-equivalent + Knapp/Bump deepening).
Steinberg formula / Kostant multiplicity	—	△	Gap (low — Master deepening). FH §25; no Codex unit. Candidate: §Master extension to 07.06.07.
Compact Lie group representation	07.07.01 compact-lie-group-representation	✓
Peter-Weyl theorem	07.07.02 peter-weyl-theorem	✓
Haar measure	07.07.03 haar-measure	✓
Weyl integration formula	—	✗	Gap (medium). FH §26.2; cornerstone of compact-Lie character theory. Candidate: 07.07.04 weyl-integration-formula.
Classical groups $\mathrm{SU}(n)$, $\mathrm{SO}(n)$, $\mathrm{Sp}(n)$ as compact-real-forms	—	△	Gap (medium). Mentioned in passing; FH §23 explicit. Could fold into a §Master extension of 07.07.01.
${\mathfrak{sp}}_{2n}$ representations	—	✗	Gap (medium). FH §16. No dedicated unit. Candidate: §Master extension of 07.06.12/07.06.05 or new unit 07.06.13.
${\mathfrak{so}}_n$ representations (B and D series)	—	△	Gap (medium). FH §§18–19. Dimension formulae implicit in 07.06.08; no worked construction.
Spin and pin representations; Clifford construction	03.09.02 clifford-algebra, 03.09.03 spin-group, 03.09.04 spin-structure, 03.09.05 spinor-bundle	△	Spin geometry chapter has the Clifford/Pin/Spin objects; representation-theoretic spin reps of ${\mathfrak{so}}_n$ (FH §20 algebraic treatment) is partial — the geometric side is in 03.09 but the algebraic half-spin construction over $\mathbb{C}$ is not its own unit. Candidate: §Master extension of 07.06.12 or 07.06.13 cross-linking to 03.09.03.
Exceptional Lie algebras $G_2,F_4,E_6,E_7,E_8$	07.04.01 mentions all five	△	Gap (low). FH §22 constructs $G_2$ explicitly; Codex states but does not construct. Master deepening only.
Symmetric functions (App B)	—	✗	Gap (low). Touched implicitly via Schur polynomials in 07.05.02. Master deepening to 07.05.04 would suffice.
Multilinear algebra refresher (App A)	01.01.* linear algebra strand	✓	Linear-algebra strand 01.01 covers tensor and exterior powers.
Lie's, Engel's, Jordan decomposition (App E)	—	△	Gap (low). Engel/Lie are prereqs for the Cartan classification; touched in 07.04.01 but no dedicated unit. Master deepening candidate.

Aggregate coverage estimate (REDUCED audit basis).

• Finite-group half of FH (Part I, §§1–6): ~80% covered. Gaps are worked $S_n$ character tables, the standalone Frobenius character formula unit, and the dedicated Schur-Weyl duality / Schur functor unit.
• ${\mathfrak{sl}}_2$ / ${\mathfrak{sl}}_3$ pedagogical centrepiece (Part II, §§7–13): ~40% covered. Codex has the abstract machinery (highest-weight, Verma modules, characters) but lacks the worked-out ${\mathfrak{sl}}_2$ and ${\mathfrak{sl}}_3$ examples that are FH's defining contribution. This is the largest pedagogical gap.
• Classical Lie algebras case-by-case (Part III, §§14–20): ~60% covered abstractly via 07.06.* (Dynkin diagrams, Weyl group, etc.) but the worked ${\mathfrak{sp}}_{2n}$, ${\mathfrak{so}}_{2n+1}$, ${\mathfrak{so}}_{2n}$ constructions are not present as their own units. Spin reps cross to 03.09 but the algebraic half-spin construction is missing.
• General theory (Part IV, §§21–26): ~85% covered. Cartan-Weyl, Weyl character formula, Weyl dimension formula, Borel- Weil all shipped. Missing: Weyl integration formula, Steinberg/Kostant multiplicities (master deepening).

Overall: ~65% of FH covered by the 26 shipped units of 07-representation-theory/ (plus spin-geometry cross-references). The gap is pedagogical, not topical — Codex has nearly all of FH's theorems but is missing the worked-example units that make FH distinctive. This is exactly the pattern the stub task predicted ("deepening-heavy outcome rather than new-unit-heavy").

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§3 Gap punch-list (priority-ordered)

Priority 1 — high-leverage, captures FH's distinctive pedagogical content:

1. 07.06.11 Representations of ${\mathfrak{sl}}_2\mathbb{C}$. Standalone worked-example unit. The $V_n={\mathrm{Sym}}^n({\mathbb{C}}^2)$ classification, weight decomposition, ladder operators $H,E,F$, character ${\chi }_n(t)=(t^{n+1}-t^{-n-1})/(t-t^{-1})$, Casimir element computed explicitly, tensor product Clebsch-Gordan rule $V_m\otimes V_n={⨁}_{k=0}^{\min (m,n)}{V}_{m+n-2k}$. FH §11 anchor; Hall §4 anchor; Hall Quantum Theory §17 anchor for the physics cross-link to angular momentum. Three-tier; this is the single most quoted unit-worth-of-material in physics-adjacent rep theory. ~2000 words. Foundational — pulls weight for 06-lie-algebraic/ and for cross-strand QM bridges.

2. 07.06.12 Representations of ${\mathfrak{sl}}_3\mathbb{C}$. The hexagonal-weight-picture unit. Cartan subalgebra of diagonal trace-zero matrices, root system $A_2$ as a planar hexagon, the six positive/negative roots, the two fundamental weights ${\omega }_1,{\omega }_2$, the standard $V={\mathbb{C}}^3$ rep and its dual, the adjoint rep (the 8 of QCD), the Clebsch-Gordan decomposition $V \otimes V^* = V_{\mathrm{ad}} \oplus V_{\mathrm{triv}}$,andaworked$V_{a, b} = \Gamma_{a \omega_1 + b \omega_2}$forsmall$(a, b)$. FH §§11–13 anchor; this is FH's pedagogical centrepiece and is the single largest gap. Three-tier; master tier includes the dimension formula via the hexagonal-Weyl count. ~2500 words. High pedagogical leverage; closes a worked- example gap that no other Codex unit fills.

3. 07.05.04 Schur-Weyl duality. Standalone unit. $\mathrm{GL}d \times S_n$actingon$V^{\otimes n}$; the bimodule decomposition $V^{\otimes n} = \bigoplus{\lambda \vdash n,, \ell(\lambda) \leq d} \mathbb{S}\lambda(V) \otimes V\lambda$where$\mathbb{S}\lambda$ is the Schur functor and $V\lambda$ is the Specht module; proof via double-centraliser theorem. FH §6 anchor; Sagan §2 anchor. Originator-prose section citing Schur 1901 and Weyl 1925. Three-tier; master tier includes the Brauer-algebra generalisation to $\mathrm{O}(V)$ and $\mathrm{Sp}(V)$. High — currently stated piecemeal across three units, no dedicated home.

Priority 2 — fills medium-priority FH content with new units:

4. 07.06.13 Representations of the classical Lie algebras ${\mathfrak{sp}}_{2n}$, ${\mathfrak{so}}_{2n+1}$, ${\mathfrak{so}}_{2n}$. One unit covering all three series via Dynkin diagrams $C_n,B_n,D_n$. Standard representations, exterior-power decompositions (${\Lambda }^{k}V$ for $\mathfrak{sp}$ is not irreducible — this is the canonical FH worked example), fundamental weights, dimension formulae for small $n$. FH §§16, 18, 19 anchor. Medium — these are FH's case-by-case constructions; Codex has the Dynkin classification but not the worked reps.

5. 07.05.05 Schur functor and polynomial representations of ${\mathrm{GL}}_n$. Definition of ${\mathbb{S}}_{\lambda }$ via Young symmetrisers; dimension formula via Schur polynomials and the Jacobi-Trudi identity; statement of the equivalence between polynomial reps of ${\mathrm{GL}}_n$ and representations of $S_n$ factored through Young symmetrisers. FH §6 + Appendix C anchor. Intermediate + Master tiers; depends on 07.05.04. Medium.

6. 07.07.04 Weyl integration formula. $\int_G f , d\mu_G = \frac{1}{|W|} \int_T f|_T \cdot |\Delta|^2 , d\mu_T$for$G$ compact connected with maximal torus $T$ and Weyl group $W$. Statement, proof sketch, and the canonical application: derivation of the Weyl character formula from the orthogonality of characters on $G$. FH §26.2 anchor. Medium — Codex has the algebraic Weyl character formula but lacks its compact-group integration derivation.

7. 07.06.10 Casimir element. Short unit (~1200 words). For a semisimple Lie algebra $\mathfrak{g}$ with Killing form $\kappa$, the Casimir $\Omega =\sum X_i{X}^i\in U(\mathfrak{g})$ where $\{X_i\}$ and $\{X^i\}$ are dual bases; $\Omega$ is central; acts as a scalar on each irreducible rep ($\Omega|{V\lambda} = \langle \lambda, \lambda + 2\rho\rangle$). FH §6 + §14 anchor. Intermediate + Master. Low-medium — currently referenced in passing in 07.06.07; a dedicated 1200-word unit cleans this up.

Priority 3 — Master-tier deepenings (not strictly required for FT equivalence, but close the gap to ≥95%):

8. §Master deepening of 07.05.02 and/or 07.05.04: Frobenius character formula for $S_n$. Full statement and FH-style proof from orthogonality of class functions. Currently only referenced. ~600 words added.

9. §Master deepening of 07.07.01: classical compact real forms $\mathrm{SU}(n)$, $\mathrm{SO}(n)$, $\mathrm{Sp}(n)$ as compact-real -forms of their complexifications. ~500 words added; cross-links to 07.06.13.

10. §Master deepening of 07.06.12/07.06.05: Pin and Spin half-spin representations of ${\mathfrak{so}}_n$ via Clifford algebras. ~700 words added; cross-references 03.09.02, 03.09.03. FH §20 anchor.

11. §Master deepening of 07.06.07: Steinberg multiplicity formula and Kostant multiplicity formula. ~400 words added. FH §25.

12. §Master deepening of 07.04.01: explicit construction of $G_2$ via octonions or the seven-dimensional Cayley algebra. ~500 words added. FH §22.

13. §Master deepening of 07.05.05: Littlewood-Richardson rule (statement + worked example, proof referred to Macdonald). ~400 words added. FH App A.

Priority 4 — survey / exercise pack follow-ups (optional):

14. Character tables: $S_3$, $S_4$, $S_5$, $A_4$, $A_5$. Exercise pack addition rather than new units. Could be folded into 07.05.01 or a new exercises-only file 07.05.E1. FH §§2, 3 anchor. ~5 worked tables.

15. §Pointer unit for Mackey's restriction-induction formula. FH §3 + §6. Could be a §Master extension to 07.01.07 rather than a new unit.

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§4 Implementation sketch (P3 → P4)

Realistic production estimate (mirroring earlier Brown-Higgins-Sivera, Lawson-Michelsohn, Bott-Tu batches):

• Priority 1 (3 new units): ~3.5 hours each = ~10–11 hours. 07.06.11 sl2, 07.06.12 sl3, 07.05.04 schur-weyl-duality.
• Priority 2 (4 new units): ~3 hours each = ~12 hours.
• Priority 3 (6 master deepenings): ~45 min each = ~4–5 hours.
• Priority 4 (exercise pack additions): ~2 hours.

Total: ~28–30 hours of focused production for full FT-equivalence coverage of FH. Fits a 4–5 day window. Priority 1 alone (~11 hours, 1.5 days) would close the largest pedagogical gap and raise effective coverage from ~65% to ~85%.

Originator-prose targets. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, originator-prose treatment with primary-source citations should appear in:

• Schur 1901 — Issai Schur's doctoral dissertation Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen (Friedrich-Wilhelms-Universität Berlin 1901). Originates the ${\mathrm{GL}}_n$-$S_n$ centraliser duality. Cite in 07.05.04.
• Frobenius 1896 — Ferdinand Georg Frobenius, Über Gruppencharaktere, Sitzungsberichte Preuß. Akad. Wiss. Berlin (1896) 985–1021; Über die Primfaktoren der Gruppendeterminante, ibid. (1896) 1343–1382. Originates the entire character-theory programme. Cite in 07.01.03, 07.01.04 (Master section).
• Weyl 1925/1926 — Hermann Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen, I–IV," Math. Z. 23 (1925) 271–309, Math. Z. 24 (1925) 328–376, 377–395, 789–791. Founds compact-Lie character theory; Weyl character formula and integration formula. Cite in 07.06.07, 07.07.04.
• Cartan 1894 — Élie Cartan, Sur la structure des groupes de transformations finis et continus, thèse, Paris 1894. Classifies simple complex Lie algebras. Cite in 07.04.01.
• Killing 1888–1890 — Wilhelm Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen, I–IV," Math. Ann. 31, 33, 34, 36 (1888–1890). The actual originator of the classification programme (Cartan corrected and completed Killing's work). Cite alongside Cartan in 07.04.01 Master section.
• Young 1900 — Alfred Young, "On Quantitative Substitutional Analysis," series of papers Proc. London Math. Soc. 33 onward (1900–1934). Originates Young symmetrisers and the tableau combinatorics. Cite in 07.05.02, 07.05.03.

Notation crosswalk. FH uses $V_{\lambda }$ for irreps of $S_n$ indexed by partitions $\lambda ⊢n$, ${\Gamma }_a$ for irreps of ${\mathfrak{sl}}_2$ of highest weight $a$, and ${\Gamma }_{(a,b)}$ for irreps of ${\mathfrak{sl}}_3$ of highest weight $a{\omega }_1+b{\omega }_2$. Codex uses $L(\lambda )$ or $V(\lambda )$ for the highest-weight rep with highest weight $\lambda$ (per 07.03.01, 07.06.06). The new units 07.06.11, 07.06.12 should adopt $V_n$ / $V_{a,b}$ for the explicit-example tier (matching FH and matching physics-strand $V_j$ angular-momentum-irrep convention) and explicitly note the $L(\lambda )$ correspondence in a §Notation paragraph.

Cross-strand weaving (Pass-W). New units should link laterally to:

• physics/quantum-mechanics/ angular momentum strand — 07.06.11 is the direct algebraic underpinning of QM angular momentum ($J^2,J_z$, raising/lowering operators).
• 02.02 quantum-theory-groups-representations audit (Woit, FT 2.02) — 07.06.11, 07.06.12, 07.07.04 are explicit prereqs for Woit's spin and $\mathrm{SU}(3)$ chapters.
• 03.09 spin-geometry — 07.06.13 (or its master deepening on spin reps) cross-links to 03.09.02, 03.09.03, 03.09.04, 03.09.05.
• 04 algebraic-geometry/Grassmannian (if/when shipped) — 07.05.05 Schur functor and Schubert calculus.

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§5 What this plan does NOT cover

• A line-number-level inventory of every named theorem in FH (full P1 audit; deferred until a local PDF is acquired). Coverage estimates in §2 are based on FH's well-known chapter structure plus spot-checks of Codex units.
• Serre's three slim books (FT 3.12 Trees, 3.13 A Course in Arithmetic — Linear Reps fragment, 3.15 Linear Representations of Finite Groups). Per the stub spec, those are deferred to their own dedicated audits. Note that FH §§1–5 substantially supersedes Serre 3.15 in pedagogical scope; the Serre 3.15 audit is therefore expected to be a thin "Serre is FH §§1–5 condensed" plan rather than a full punch-list.
• Diaconis 3.16 Probability and Representation Theory — own audit; FH does not cover the probability-of-random-walks application.
• Hall Lie Groups, Lie Algebras, and Representations (FT 3.10 -adjacent) — own audit; significant overlap with FH Part II but matrix-Lie-group-first ordering rather than Lie-algebra-first.
• The infinite-dimensional rep theory of non-compact Lie groups (Harish-Chandra, Mackey, Kirillov orbit method). FH excludes this by design; the Codex follows FH.
• Modular representation theory (Brauer characters, blocks, decomposition matrices). FH excludes; Codex follows.
• Algebraic groups in positive characteristic (Jantzen Representations of Algebraic Groups; Lusztig; quantum groups). FH excludes; deferred to a hypothetical future Master-tier algebraic-groups audit.
• Hopf-algebraic and category-theoretic abstractions (tensor categories, fusion rules, modular tensor categories). FH excludes; deferred to the planned 05-category-theory/ strand.

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§6 Acceptance criteria for FT equivalence (FH)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4, the book is at equivalence-coverage when:

• The 3 Priority-1 new units have shipped (07.06.11 sl2, 07.06.12 sl3, 07.05.04 schur-weyl-duality). These close the largest pedagogical gap.
• ≥95% of FH's named theorems and major worked examples in chapters 1–26 map to a Codex unit (currently ~65%; after Priority-1 this rises to ~85%; after Priority-1+2 to ~92%; full ≥95% requires Priority-3 master deepenings).
• ≥90% of FH's worked computations have a direct unit or are cross-referenced from a unit that covers them. The ${\mathfrak{sl}}_2$, ${\mathfrak{sl}}_3$, ${\mathfrak{sp}}_{2n}$, ${\mathfrak{so}}_n$ case studies all need dedicated units or §Master extensions.
• Notation crosswalk recorded (see §4).
• Pass-W weaving connects the new units to physics/quantum-mechanics/, to the Woit audit, and to 03.09 spin-geometry/.
• Originator-prose sections (Schur 1901, Frobenius 1896, Weyl 1925, Cartan 1894, Killing 1888, Young 1900) appear in the relevant Master tiers.

Equivalence verification protocol (per FASTTRACK_EQUIVALENCE_PLAN.md §4):

• 4.1 Book-as-input self-check. Sample 5 random theorem statements from FH; for each, identify the Codex unit that proves it.
• 4.2 Exercise reproducibility. Sample 5 random exercises from FH (likely from §§3, 4, 11, 13, 22); for each, identify the Codex unit(s) sufficient to solve it. Note: FH exercises are notoriously open-ended and often constitute their own mini-papers — the sampling protocol should bias toward computational rather than open-ended exercises.
• 4.3 Notation comprehension. Sample 5 fragments of FH prose; paraphrase each into the Codex notation system (per §4 crosswalk).

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§7 Sourcing

• Local PDF status. Not present in reference/textbooks-extra/ (verified — only the cover image reference/fast-track/images/Fulton-Harris-Rep-Theory-683x1024__3816f354c4.jpg is local). FH is a commercial Springer GTM and is not author-hosted.
• Commercial source. Springer GTM 129. Print + Springer eBook available at link.springer.com/book/10.1007/978-1-4612-0979-9. ISBN 0-387-97527-6 (hardcover), 0-387-97495-4 (softcover).
• Library-mirror sources. Acquire via institutional access (university library Springer subscription) or interlibrary loan. For the local copy, target placement is reference/textbooks-extra/Fulton-Harris-Representation-Theory.pdf to mirror the pattern of other commercial-source FT texts.
• Companion / supplementary materials.
  • W. Fulton, Young Tableaux: With Applications to Representation Theory and Geometry (Cambridge LMS Student Texts 35, 1997) — the standalone expansion of FH §4. Public eBook via Cambridge Core.
  • I. G. Macdonald, Symmetric Functions and Hall Polynomials (Oxford 2nd ed. 1995) — referenced by FH for the full Littlewood-Richardson proof.
• Open-access alternatives covering ~60% of FH:
  • B. C. Hall, Lie Groups, Lie Algebras, and Representations (Springer GTM 222) — substantial preview on Google Books; covers FH Part II at gentler pace.
  • B. E. Sagan, The Symmetric Group (Springer GTM 203) — covers FH §§1–6 finite-group half.
  • Various lecture notes: Etingof et al. Introduction to Representation Theory (free, MIT 18.712 notes; covers FH §§1–6 with cleaner category-theoretic spine).
• Reduced-audit flag. This plan was produced without a local FH PDF. Coverage estimates in §2 should be re-verified once a PDF is acquired, particularly for the Part III chapters on classical Lie algebras (FH §§14–20) where worked-example detail is the load-bearing pedagogical content. Promote to full P1 audit when PDF is local.