Serre — Linear Representations of Finite Groups (Fast Track 3.15) — Audit + Gap Plan

Book: Jean-Pierre Serre, Linear Representations of Finite Groups (Springer Graduate Texts in Mathematics 42, English translation by Leonard L. Scott 1977 of the French original Représentations linéaires des groupes finis, Hermann 1967; 2nd French ed. Hermann 1971; ≈ x + 170 pp.). ISBN 0-387-90190-6 (hardcover), 0-387-90190-6 (softcover reprint). Commercial title (BUY in docs/catalogs/FASTTRACK_BOOKLIST.md row 3.15).

Fast Track entry: 3.15. The canonical shortest graduate monograph on finite-group representation theory. Distinguished from the Fast Track's other rep-theory slots (3.10 Hall; 3.11 Fulton-Harris; 3.12 Serre CSLA; 3.13 Serre Lie Algebras and Lie Groups; 3.16 Diaconis) by being the slim crystalline finite-group treatment that Fulton-Harris §§1–5 "fattens" into a 100-page worked-example tour. Three-part structure: Part I (Chs. 1–5) character theory of finite groups over $C$ ; Part II (Chs. 6–11) representations in characteristic $0$ — induced representations, Mackey's irreducibility criterion, Artin's and Brauer's induction theorems, rationality, the Schur indicator; Part III (Chs. 12–18) modular representations — Grothendieck groups $R_{K} (G), R_{k} (G), P_{k} (G)$ , the cde triangle, Brauer characters, decomposition and Cartan matrices, blocks. Part III is the foundational short-form treatment of Brauer's modular theory and has no Fast Track analogue elsewhere.

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 + new units so that Linear Representations of Finite Groups (LRFG 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/ (verified — directory listing confirms no Serre LRFG; only the cover image reference/fast-track/images/Serre-Lie-Algebras-Lie-Groups-712x1024__dd9fad6ce3.jpg is local, and the Fast Track source page reuses that cover image across multiple Serre slots — it does not correspond to LRFG). WebFetch on Springer link (link.springer.com/book/10.1007/978-1-4684-9458-7) is gated; Anna's Archive search yields hosted copies but acquisition is deferred to a follow-up sourcing pass. This audit works from (a) the well-documented public TOC of LRFG (three-part structure as above, referenced verbatim in 07.01.07 exercise notes — "Serre §7.4", "Serre, Linear Representations of Finite Groups Part III (Chapters 9–10)"); (b) the Codex's 26 shipped 07-representation-theory/ units, of which 9 already cite Serre LRFG by name in their tier-anchor / reference blocks; (c) the canonical secondary literature (Curtis-Reiner, Isaacs, Alperin, James-Liebeck). 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 and Fulton-Harris 3.11.

§1 What LRFG is for

LRFG is the canonical short graduate text on finite-group representation theory. Where Fulton-Harris §§1–5 (FT 3.11) gives the same Part-I content as the warmup chapter of a 550-page Lie-algebra book, where Curtis-Reiner Methods of Representation Theory (Vols. I–II, 1981/87) is the encyclopaedic 2000-page reference, where Isaacs Character Theory of Finite Groups (1976) is the 300-page character-theory monograph at the same level, where James-Liebeck Representations and Characters of Groups (1993/2001) is the gentle undergraduate-flavoured exposition, and where Alperin Local Representation Theory (Cambridge Studies in Advanced Mathematics 11, 1986) gives the modern modular treatment via local methods (Brauer pairs, defect groups, Green correspondence) — LRFG is the shortest rigorous self-contained graduate textbook covering both the ordinary and modular theory in 170 pages. Like CSLA (FT 3.12), LRFG extracts a lecture course (Serre, École Normale Supérieure 1962, ENSJF 1966) into its functorial spine, stripping commentary while keeping every non-trivial proof. The book is the standard reference for finite-group character theory among working algebraists, number theorists (Artin $L$ -functions, Galois representations), and group-theoretic physicists.

Distinctive contributions, in the order LRFG develops them:

Part I: Character theory over $C$ (Chs. 1–5, ~45 pp.). Definitions, Maschke's theorem, Schur's lemma, characters and orthogonality, the regular representation, the canonical decomposition into isotypic components, induced representations, group of a product, tensor products. Serre's signature economy: the canonical decomposition is proved in 2 pages; the orthogonality relations are derived as immediate consequences of Schur's lemma applied to $Hom_{G} (V, W)$ . The whole of finite-group character theory over $C$ is in 45 pages.
Worked character tables (Ch. 5, ~10 pp.). Serre works out the character tables of $C_{n}$ , $D_{n}$ , $A_{4}$ , $S_{4}$ , $A_{5}$ , and (via §5.8) the Heisenberg group, in a sequence of $\approx$ 1-page computations. These canonical small-group tables are the load-bearing examples cited throughout the literature; they are notably absent from Fulton-Harris (which gives only $S_3, S_4, S_5$). This concrete-table material is the most-quoted-by-physicists part of LRFG.
Part II: Subgroup representations (Chs. 6–11, ~50 pp.). Induced representations, Frobenius reciprocity, Mackey's theorem and the irreducibility criterion (Ch. 7), examples (Ch. 8: $S_n, A_n, GL_2(\mathbb{F}_q)$), Artin's induction theorem (Ch. 9), Brauer's induction theorem (Ch. 10), rationality and the Schur index (Ch. 11–12). Mackey's irreducibility criterion (§7.4) and the Artin–Brauer induction theorems (Chs. 9–10) are the load-bearing theorems here — they have no analogous unit in the current Codex, despite being load-bearing for number-theoretic applications (Artin $L$ -functions, the Brauer-Heller-Hasse conjecture).
Artin's induction theorem (Ch. 9). Every character of $G$ is a $Q$ -linear combination of characters induced from cyclic subgroups. Originator: Emil Artin, Über eine neue Art von L-Reihen, Abh. Math. Sem. Univ. Hamburg 3 (1923); refined in Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren, Abh. Math. Sem. Univ. Hamburg 8 (1930). The theorem's number-theoretic import: every Artin $L$ -function is a $Q$ -linear combination of Hecke $L$ -functions, but the integrality required for analytic continuation is what Brauer 1946 provides.
Brauer's induction theorem (Ch. 10). Every character of $G$ is a $Z$ -linear combination of characters induced from elementary subgroups (subgroups of the form $C \times P$ where $C$ is cyclic of order coprime to a prime $p$ , $P$ is a $p$ -group). Originator: Richard Brauer, On Artin's L-series with general group characters, Ann. Math. 48 (1947). This is the deep result powering Artin's conjecture on the holomorphy of $L$ -functions; the $Z$ -linearity (vs. Artin's $Q$ -linearity) makes the difference between meromorphy and holomorphy modulo abelian parts.
Part III: Modular representation theory (Chs. 12–18, ~50 pp.). Serre's distinctive contribution and the part with no Fast Track analogue. Representations of $G$ over a modular field $k$ of characteristic $p ∣ ∣ G ∣$ , where Maschke fails. The three Grothendieck groups $R_{K} (G), R_{k} (G), P_{k} (G)$ ( $K$ the field of characteristic 0, $k$ residue, $K$ -irreducible characters, modular characters, projective indecomposable characters), connected by the cde-triangle: $c : P_{k} (G) \to R_{k} (G)$ (Cartan map), $d: R_K(G) \to R_k(G) $(d eco m p os i t i o nma p),$ e: P_k(G) \to R_K(G)$ (lift of projectives). Brauer characters; the Cartan matrix $C = d \circ e^{t}$ ; the decomposition matrix $D$ relating $K$ -characters to modular characters; the Fong-Swan theorem for $p$ -solvable groups; blocks and the Brauer first/second/third main theorems are stated.
The cde-triangle and Brauer's theorem on modular characters (Chs. 14–16). Brauer's theorem: every irreducible modular character of a finite group $G$ in characteristic $p$ lifts to an ordinary character of an elementary subgroup, induced up. This is the modular analogue of Brauer's ordinary induction theorem and the foundation of Brauer's modular representation theory. Originator: Richard Brauer, sequence of papers 1935–1956 in Math. Ann., Math. Z., Trans. AMS, culminating in Number theoretical investigations on groups of finite order, Proc. Int. Symp. Tokyo-Nikko 1955.
Blocks (Ch. 17–18). Definition of a block as an indecomposable two-sided ideal of $k G$ ; defect groups; the Brauer first main theorem (bijection between blocks of $G$ with defect group $D$ and blocks of $N_{G} (D)$ with defect group $D$ ). Serre states the main theorems and proves the easier ones (the first; the second is stated only). This material is the gateway to the modern theory (Alperin's weight conjecture, Broué's abelian-defect conjecture, the McKay conjecture — all referenced from 07.02.01-maschke-theorem Master tier but with no anchor unit).

LRFG is not the place to learn Lie theory (Serre 3.12 CSLA and 3.13 Lie Algebras and Lie Groups are siblings), is not the place for exhaustive worked examples in the Fulton-Harris sense (LRFG's worked character tables are minimal and crystal-clear, not pedagogical), and is not the place for the deep modular structure theory of Curtis-Reiner Vol. II or Alperin's local methods (LRFG Part III is a foundational gateway to that theory, not a comprehensive treatment). The canonical follow-ups are Curtis-Reiner Vols. I–II for the encyclopaedic ordinary + modular treatment; Isaacs for character theory in depth; Alperin Local Representation Theory for the modular structure theory; Navarro Character Theory and the McKay Conjecture (2018) for the modern state of the art.

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

C. W. Curtis, I. Reiner, Methods of Representation Theory, with Applications to Finite Groups and Orders, Vols. I (1981) and II (1987) (Wiley-Interscience / AMS Chelsea reprint). The canonical encyclopaedic reference; Vol. I covers ordinary representation theory (corresponding to LRFG Parts I–II), Vol. II covers modular and integral representation theory (LRFG Part III + much more). Referenced 8 times in shipped Codex units (07.01.05, 07.01.07, 07.01.08, 07.02.01).
I. M. Isaacs, Character Theory of Finite Groups (Academic Press 1976; AMS Chelsea reprint 2006). The 300-page character-theory monograph at the same level as LRFG Parts I–II, with substantially more depth on Frobenius groups, $M$ -groups, $p$ -solvable groups, and the Brauer-Suzuki theorem. The Fast Track analogue / depth-deepening pair for LRFG Parts I–II.
W. Fulton, J. Harris, Representation Theory: A First Course (Springer GTM 129, 1991) — Fast Track 3.11. Parts I (§§1–6) cover the same ordinary-character content as LRFG Parts I–II, "fattened" into a 100 page worked-example tour. Already audited in plans/fasttrack/fulton-harris-representation-theory.md; that audit observes "Serre 3.15 is FH §§1–5 condensed" and defers the LRFG stub. This plan completes that deferred stub.
J. L. Alperin, Local Representation Theory: Modular Representations as an Introduction to the Local Representation Theory of Finite Groups (Cambridge Studies in Advanced Mathematics 11, Cambridge University Press 1986). The canonical short modern monograph on Brauer's modular theory via local methods (Brauer pairs, defect groups, Green correspondence). The Fast-Track-style follow-up to LRFG Part III; its techniques are the modern toolkit (decomposition matrices, blocks, Green correspondence) for which LRFG Part III is the introductory text.
(Optional fifth peer source.) G. James, M. Liebeck, Representations and Characters of Groups (Cambridge University Press, 2nd ed. 2001). The gentle undergraduate-flavoured exposition of LRFG Part I; cited 3 times in shipped units (07.01.03, 07.01.04, 07.01.05).

§2 Coverage table (Codex vs LRFG)

Cross-referenced against the current 26 shipped units of content/07-representation-theory/ (8 in 01-foundations/, 1 in 02-character/, 1 in 03-highest-weight/, 1 in 04-classification/, 3 in 05-symmetric/, 9 in 06-lie-algebraic/, 3 in 07-compact-lie/). ✓ = covered, △ = partial / different framing / present only as commentary or exercise, ✗ = not covered.

LRFG topic	Chapter	Codex unit(s)	Status	Note
PART I — ORDINARY CHARACTERS
Group representation, equivalence, sub/quotient	Ch. 1	`07.01.01`	✓	Master tier cites Serre §1 as anchor.
Maschke's theorem (complete reducibility, char 0)	Ch. 1	`07.02.01`	✓	Maschke 1899 originator-prose section present.
Schur's lemma	Ch. 2	`07.01.02`	✓	Master tier cites Serre §2.2.
Character of a representation	Ch. 2	`07.01.03`	✓	Master tier cites Serre §2; Frobenius 1896 originator-prose.
Character orthogonality relations (first and second)	Ch. 2	`07.01.04`	✓	Master tier cites Serre §2; full proofs via Schur.
Canonical decomposition into isotypic components	Ch. 2	△ (stated in `07.01.01` and `07.01.05`)	△	Gap (low). No standalone unit; Serre §2.6 gives the explicit projector formula $p_i = (n_i/
Regular representation	Ch. 2	`07.01.05`	✓	Artin-Wedderburn decomposition derived.
Number of irreducibles = number of conjugacy classes	Ch. 2	△ (in `07.01.04` Master tier)	△	Gap (low). Standard fact, but no dedicated unit. Candidate: §Intermediate extension to `07.01.04`.
Tensor product of representations	Ch. 3	`07.01.06`	✓
Group of a product $G \times H$ ; reps as tensors	Ch. 3	—	✗	Gap (low). Serre §3.2; cleanly stated theorem $G \times H = G \otimes H$ . Candidate: §Master extension to `07.01.06`.
Symmetric and alternating squares; $χ_{Sym^{2}}, χ_{Λ^{2}}$ formulae	Ch. 2	△ (in `07.01.06`)	△	Gap (low). Touched but explicit formulae missing. Master deepening.
Worked character tables: $C_{n}, D_{n}, A_{4}, S_{4}, A_{5}, Q_{8},$ Heisenberg	Ch. 5	—	✗	Gap (medium — defining LRFG content). No Codex unit. These tables are the most-quoted material in LRFG; their absence is a real gap that the Fulton-Harris audit also flagged. Candidate: dedicated `07.02.E1` exercise/worked-tables file (5–6 tables, ~1500 words).
PART II — INDUCED REPS AND INDUCTION THEOREMS
Induced representation $Ind_{H}^{G}$	Ch. 7	`07.01.07`	✓	Comprehensive unit; cites Serre §7.
Frobenius reciprocity	Ch. 7	`07.01.08`	✓	Categorical + character forms.
Induction in stages $Ind_{H}^{G} = Ind_{K}^{G} \circ Ind_{H}^{K}$	Ch. 7	△ (in `07.01.07` Exercise 2)	△	Gap (low). Proved as exercise; could be elevated.
Mackey's decomposition formula ( $Res_{K}^{G} \circ Ind_{H}^{G} W$ over double cosets)	Ch. 7	△ (in `07.01.07` Exercise 6)	△	Gap (medium). Present as Exercise 6 of `07.01.07` with full statement and proof sketch, but no standalone Mackey-theory unit. Candidate: promote to new unit `07.01.09 mackey-decomposition-formula`.
Mackey's irreducibility criterion	Ch. 7.4	△ (commentary in `07.01.07` line 410)	△	Gap (medium-high). Stated only as commentary at the end of `07.01.07`; no formal theorem-of-record, no proof. Load-bearing for Part II. Candidate: §Master section of the new `07.01.09` unit.
Worked examples: $S_{n}, A_{n}, G L_{2} (F_{q})$ via induction	Ch. 8	△ (partial in `07.05.01`)	△	Gap (medium). $S_{n}$ covered via Young symmetrisers (`07.05.01–03`) but not via Serre's induction-from-Young-subgroups approach; $G L_{2} (F_{q})$ entirely absent. Candidate: dedicated worked-example unit.
Artin's induction theorem	Ch. 9	△ (commentary in `07.01.07`, `07.01.01`)	△	Gap (high — anchor missing). Brauer's theorem is mentioned in `07.01.07` line 412 and `07.01.01` line 377 but Artin's $Q$ -linear version is not explicitly stated anywhere. The two are usually paired. Candidate: new unit `07.01.10 artin-induction-theorem`.
Brauer's induction theorem	Ch. 10	△ (commentary only in `07.01.07` line 412, `07.01.01` line 377, `07.01.03` line 405)	△	Gap (high — anchor missing). Stated but never proved; no dedicated unit. The depth of the theorem (integrality via algebraic-integer arguments) is acknowledged but not exposed. Candidate: new unit `07.01.11 brauer-induction-theorem`.
Rationality, the Schur index	Ch. 12	—	✗	Gap (medium-low). Not in Codex. The Schur index $m (χ)$ measures how far a $Q$ -irreducible character is from being $Q$ -representation-realisable. Candidate: short Master-tier unit.
Frobenius-Schur indicator	Ch. 13	—	✗	Gap (medium). Not in Codex. The FS indicator $ν (χ) \in {0, + 1, - 1}$ classifies irreducibles into real / quaternionic / complex types; load-bearing for the unitary-group / orthogonal-group / symplectic-group cross-link. Candidate: new unit `07.01.12 frobenius-schur-indicator`.
PART III — MODULAR REPRESENTATIONS
Modular representations over $k$ , $\mathrm{char}, k = p \mid	G	$	Ch. 14	△ (commentary in `07.01.01`, `07.02.01`, `07.01.03`, `07.05.03`)
Grothendieck groups $R_{K} (G), R_{k} (G), P_{k} (G)$	Ch. 14	—	✗	Gap (high). The cde-triangle organisational backbone of LRFG Part III. No Codex unit. Candidate: `07.02.03 grothendieck-group-cde-triangle`.
Decomposition map $d : R_{K} (G) \to R_{k} (G)$ and decomposition matrix $D$	Ch. 15	△ (commentary in `07.02.01`, `07.05.03`)	△	Gap (high). Decomposition matrices are referenced as exposition (`07.02.01` Master tier; `07.05.03` for $S_{n}$ specifically) but no anchor unit defining the construction. Candidate: §Master section of `07.02.03` or new unit.
Cartan map $c : P_{k} (G) \to R_{k} (G)$ and Cartan matrix $C$	Ch. 15	—	✗	Gap (high). No Codex unit. Cartan matrix $C = D^{t} D$ is the second pillar of the cde-triangle. Candidate: §Master section of `07.02.03`.
Brauer characters	Ch. 17	△ (commentary in `07.01.03` line 415, `07.01.04` line 443, `07.02.01`)	△	Gap (HIGH — anchor missing). Defined only as a sentence ("characters lifted from $p$ -roots of unity to $Q$ , evaluated on $p$ -regular elements"). No formal definition, no orthogonality relations, no examples. Candidate: new unit `07.02.04 brauer-character`.
Brauer's theorem on modular characters (lifting from elementary subgroups)	Ch. 17	—	✗	Gap (high). Modular analogue of Brauer's ordinary induction theorem; no Codex unit. Candidate: §Master section of `07.02.04` or new `07.02.05 brauer-modular-induction`.
Fong-Swan theorem ( $p$ -solvable: every modular irrep lifts to char 0)	Ch. 17	—	✗	Gap (medium). Bridges modular reps of $p$ -solvable groups to ordinary reps. Candidate: pointer in `07.02.04` Master tier.
Blocks of $k G$ ; defect groups	Ch. 18	△ (commentary in `07.02.01`)	△	Gap (medium). Blocks defined only as a sentence in `07.02.01` Master tier ("decomposition of $k G$ into indecomposable two-sided ideals"); no Codex unit develops the theory. Candidate: new unit `07.02.06 block-theory`.
Brauer's first/second/third main theorems on blocks	Ch. 18	—	✗	Gap (medium — survey-level). Referenced via Alperin's-weight-conjecture commentary in `07.02.01`. Candidate: §Master section of `07.02.06`; full proofs deferred to Alperin / Curtis-Reiner Vol. II audit.

Aggregate coverage estimate (REDUCED audit basis).

Part I (LRFG Chs. 1–5, character theory over $C$ ): ~90% covered. All theorems shipped via 07.01.01–08 + 07.02.01. Gaps: worked character tables of small groups (Serre Ch. 5), the explicit canonical-decomposition projector formula, and minor details (group of a product, $Sym^{2} / Λ^{2}$ character formulae). These are pedagogical-completeness gaps, not structural.
Part II (LRFG Chs. 6–13, induced reps and induction theorems): ~50% covered. 07.01.07–08 cover induction + Frobenius reciprocity comprehensively. Mackey's decomposition formula is present as Exercise 6 of 07.01.07; Mackey's irreducibility criterion is mentioned in passing. Artin's induction theorem, Brauer's induction theorem, the Schur index, and the Frobenius-Schur indicator are NOT anchored — they appear only as forward-references in commentary blocks. This is the largest structural gap in the ordinary-theory half.
Part III (LRFG Chs. 14–18, modular representations): ~10% covered. Modular representation theory is referenced 9+ times across 5 shipped units as a forward-pointer (Brauer 1935 originator citations, the McKay conjecture, decomposition matrices) but there is no anchor unit anywhere in the Codex. The Grothendieck-group cde-triangle, Brauer characters, blocks, defect groups, and the Brauer main theorems are all completely absent. This is the single largest gap in the entire 07 chapter.

Overall: ~55% of LRFG covered by the 26 shipped units of 07-representation-theory/. The gap is structural, not pedagogical — unlike the Fulton-Harris audit (which found ~65% coverage with the gap being worked examples), the LRFG audit finds the gap concentrated in Mackey theory, Artin-Brauer induction, and the entirety of modular representations. Closing these gaps would also retroactively fix the 5 shipped units that silently depend on modular-rep concepts with no anchor (see §3 below).

Silent-dependency findings (units that forward-reference unanchored content):

07.01.01-group-representation.md line 377: states Brauer's induction theorem without anchor.
07.01.03-character.md lines 405, 415: states Brauer's theorem on induced characters and modular (Brauer) characters without anchor.
07.01.04-character-orthogonality.md line 443, 473: states modular orthogonality / Brauer 1935 without anchor.
07.01.07-induced-representation.md lines 412, 432, 458, 463–464: states Brauer's induction theorem, Mackey 1949–52, Mackey's decomposition formula (as exercise), Mackey's irreducibility criterion without anchor units; cites "Serre §7.4" and "Serre, Linear Representations of Finite Groups Part III (Chapters 9–10)" directly.
07.01.08-frobenius-reciprocity.md line: cites Mackey projection formula; Mackey theory referenced as chapter-level content of Serre Part III.
07.02.01-maschke-theorem.md (multiple lines): the entire Master tier is built around modular-rep theory as the complement of Maschke; cites Brauer 1935, decomposition matrices, blocks, defect groups, Alperin's weight conjecture, Broué's abelian-defect conjecture, McKay conjecture, Rickard equivalences — all forward pointers with no anchor.
07.05.03-specht-module.md (multiple lines): modular Specht modules, decomposition matrices for $S_{n}$ , Mullineux conjecture, Carter conjecture, Hecke algebras at roots of unity, KLR categorification — all built on a modular-rep foundation that has no Codex anchor.

This silent-dependency pattern means LRFG anchoring would resolve a lattice-wide hole in the rep-theory chapter — not just a local gap.

§3 Gap punch-list (priority-ordered)

Priority 1 — high-leverage, captures LRFG's distinctive content and resolves silent-dependency holes elsewhere:

07.02.02 Modular representation (over a $p$ -modular triple $(K, O, k)$ ). Foundational anchor. Definition of the modular setup: $k$ field of characteristic $p$ , $O$ complete DVR with residue field $k$ , $K = Frac (O)$ of characteristic $0$ ; the three group algebras $KG, \mathcal{O}G, kG $; l i f t in g an d r e d u c t i o n - m o d -$ p$ functors. Statement (without proof) of Maschke's failure in characteristic $p ∣ ∣ G ∣$ and the resulting indecomposable-vs-irreducible distinction. Three-tier; ~1800 words. LRFG §14 anchor; Alperin Chs. 1–3 anchor; Curtis-Reiner Vol. II §6 anchor. Foundational — closes the 9+ silent-dependency forward-references.
07.02.03 Grothendieck groups and the cde-triangle. The organisational spine of LRFG Part III. $R_{K} (G) =$ free abelian group on $K$ -irreducible characters, $R_{k} (G) =$ on $k$ -irreducible Brauer characters, $P_{k} (G) =$ on projective indecomposable modular characters. The maps: decomposition $d : R_{K} (G) \to R_{k} (G)$ (reduce mod $p$ ), Cartan $c : P_{k} (G) \to R_{k} (G)$ (the natural inclusion of projectives), lift $e : P_{k} (G) \to R_{K} (G)$ (extend scalars to $K$ ). The commuting triangle $c = d \circ e$ . The Cartan matrix $C = D^{t} D$ (where $D$ is the decomposition matrix); $det (C)$ is a power of $p$ . Three-tier; master tier requires the $C = D^{t} D$ derivation. ~2200 words. LRFG §§15–16 anchor. High — the cde-triangle is the second pillar of LRFG Part III and is referenced (without name) in 07.05.03 for the $S_{n}$ decomposition matrix.
07.02.04 Brauer character. Definition: for an irreducible $k G$ -module $V$ with $k = \overline{F_{p}}$ , the Brauer character $β_{V} : G_{p^{'}} \to C$ on $p$ -regular elements (those of order coprime to $p$ ) is $\beta_V(g) = \sum_i \widehat{\lambda_i} $w h er e$ {\lambda_i} $a r ee i g e n v a l u eso f$ g$ on $V$ and $\cdot$ is the Teichmüller lift to roots of unity in $\overline{Q_{p}} \subset C$ . Brauer orthogonality relations (analogue of $χ$ -orthogonality on $p$ -regular elements). The number of irreducible modular characters equals the number of $p$ -regular conjugacy classes (Brauer's theorem). Three-tier; ~2000 words. LRFG §17 anchor; Isaacs Ch. 15 anchor (modular character supplement). High — anchors the 5+ forward-references in 07.01.03, 07.01.04, 07.02.01, 07.05.03.
07.01.10 Artin's induction theorem. Every character of a finite group $G$ over $C$ is a $Q$ -linear combination of characters induced from one-dimensional characters of cyclic subgroups. Statement; proof via the integrality of the character table and a $Q$ -linear-algebra argument (Serre's elementary proof). Originator-prose section citing Artin
1. Three-tier; ~1500 words. LRFG §9 anchor. High — currently not anchored anywhere despite being a textbook-standard theorem and the pair partner to Brauer's induction theorem.
07.01.11 Brauer's induction theorem. Every character of $G$ over $C$ is a $Z$ -linear combination of characters induced from one-dimensional characters of elementary subgroups (subgroups of the form $C \times P$ , $C$ cyclic of order coprime to a prime $p$ , $P$ a $p$ -group). Statement; full proof requires ~10 pages of careful integrality-of-character-values arguments (Brauer 1946; Serre's exposition follows Brauer-Tate); the Codex unit should give the statement, the structure of the proof, and a one-page sketch of the integrality lemma. Originator- prose section citing Brauer 1946. Three-tier; ~2000 words. LRFG §10 anchor; cited 3 times in shipped Codex units as forward reference. High — load-bearing for number-theoretic applications (Artin $L$ -functions, the Brauer-Heller-Hasse conjecture).

Priority 2 — medium-priority new units / promotions of existing exercise material:

07.01.09 Mackey's decomposition formula and irreducibility criterion. Promote Mackey's decomposition formula from 07.01.07 Exercise 6 (where it currently lives with full statement + proof sketch) to a standalone unit. Add Mackey's irreducibility criterion (1951) as the key theorem of the unit. Worked example: $Ind_{H}^{G} W$ irreducible for $G = S_3, H = A_3, W = $ a non-trivial character; the non-irreducibility case via the FS-indicator-related obstruction. Three-tier; ~1800 words. LRFG §§7.3–7.4 anchor. Medium — currently buried as an exercise; standalone unit clarifies the structure and provides a citation target for 07.01.08 Master-tier "Mackey projection formula" line.
07.01.12 Frobenius-Schur indicator. Definition $\nu(\chi) = (1/|G|) \sum_{g \in G} \chi(g^2)$; classification of irreducibles over $R$ into real ( $ν = + 1$ ), quaternionic ( $ν = - 1$ ), complex ( $ν = 0$ ). Worked examples: $S_{n}$ (all real), $Q_{8}$ (one quaternionic irrep). Cross-link to the $O / U / Sp$ trichotomy on the Lie side and to spin geometry. Three-tier; ~1500 words. LRFG §13 anchor. Medium — the natural bridge between LRFG and the spin/orthogonal/symplectic real-form theory of 07.06.13 (FH §20) and 03.09.03 spin-group.
07.02.06 Block theory of $k G$ . Definition of a block as an indecomposable two-sided ideal of $k G$ ; defect groups via the $G$ -conjugacy class of $p$ -Sylow subgroups associated to the block; statement of Brauer's first main theorem (bijection between blocks of $G$ with defect group $D$ and blocks of $N_{G} (D)$ with defect group $D$ ); second and third main theorems stated only. Three-tier; master tier sketches the proof of the first main theorem via the Brauer homomorphism; ~2000 words. LRFG §18 anchor; Alperin §§13–15 anchor. Medium — the gateway to modern modular-rep theory (Alperin weight conjecture, Broué abelian defect conjecture). Resolves 07.02.01 Master-tier forward references.

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

§Master deepening of 07.01.05: canonical decomposition projector formula. Serre §2.6's explicit projector $p_i = (n_i/|G|) \sum \overline{\chi_i(g)} \rho(g)$. ~400 words added.
§Master deepening of 07.01.06: group of a product $G \times H = G \otimes H$ and the $Sym^{2} / Λ^{2}$ character formulae. ~500 words added. LRFG §3.2.
§Master deepening of 07.01.07: induction in stages (currently Exercise 2) and the projection formula $\mathrm{Ind}_H^G(W \otimes \mathrm{Res}_H^G V) = (\mathrm{Ind}_H^G W) \otimes V$. Promote to numbered theorem. ~400 words added.
§Master deepening of 07.02.04: Fong-Swan theorem (every irreducible $\overline{F_{p}} G$ -module of a $p$ -solvable group $G$ lifts to a $\overline{Q_{p}} G$ -module). ~500 words added. LRFG §17.
§Master deepening of 07.01.11: Schur index $m (χ)$ and rationality (Brauer-Heller-Speiser theorem). ~500 words added. LRFG §12.

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

07.02.E1 Worked character tables. Exercise-pack file (not a full unit) with the 6 canonical small-group tables from LRFG Ch. 5: $C_{n}$ , $D_{n}$ , $A_{4}$ , $S_{4}$ , $A_{5}$ , $Q_{8}$ , plus the Heisenberg group $H_{p}$ of order $p^{3}$ . ~2000 words. Optional — high pedagogical value; the Fulton-Harris audit independently flagged the same gap (which it deferred to its own exercise pack).
§Pointer in 07.01.10 Master tier: Artin $L$ -functions and the Artin holomorphy conjecture. ~400 words. Notes that Brauer-induction gives meromorphy and that Artin's conjecture (every non-trivial Artin $L$ -function is entire) is still open in general. Cross-links to a future number-theory/ chapter.
§Pointer in 07.02.06 Master tier: Alperin weight conjecture, Broué abelian defect conjecture, McKay conjecture. Already sketched in 07.02.01 Master tier; consolidate the canonical statements in 07.02.06. ~600 words.

§4 Implementation sketch (P3 → P4)

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

Priority 1 (5 new units): ~3.5–4 hours each = ~18–20 hours. 07.02.02 modular-representation, 07.02.03 grothendieck-cde, 07.02.04 brauer-character, 07.01.10 artin-induction-theorem, 07.01.11 brauer-induction-theorem.
Priority 2 (3 new units): ~3 hours each = ~9 hours. 07.01.09 mackey-decomposition-irreducibility, 07.01.12 frobenius-schur-indicator, 07.02.06 block-theory.
Priority 3 (5 master deepenings): ~45 min each = ~4 hours.
Priority 4 (exercise-pack additions and pointers): ~3 hours.

Total: ~34–36 hours of focused production for full FT-equivalence coverage of LRFG. Fits a 5–7 day window. Priority 1 alone (~20 hours, ~2.5 days) would close the largest structural gap — anchoring Mackey/Artin/Brauer ordinary induction theorems and establishing the Brauer-character / cde-triangle modular foundation — and would raise effective coverage from ~55% to ~85%. Priority 1 also retroactively resolves the silent-dependency findings in §2 (9+ forward-references across 5 shipped units).

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

Frobenius 1896 — Ferdinand Georg Frobenius, Über Gruppencharaktere, Sitzungsberichte Preuß. Akad. Wiss. Berlin (1896) 985–1021; Über die Primfaktoren der Gruppendeterminante, ibid. (1896) 1343–1382; Über die Charaktere endlicher Gruppen, ibid. (1896) 1023–1043. Originates character theory entirely. Already cited in 07.01.03, 07.01.04 Master tier; no new citation needed for LRFG units.
Schur 1904/1905 — Issai Schur, Neue Begründung der Theorie der Gruppencharaktere, Sitzungsberichte Preuß. Akad. Wiss. Berlin (1905) 406–432. Reproved Frobenius's character theory via the orthogonality relations as immediate consequences of Schur's lemma — the proof technique LRFG Ch. 2 inherits directly. Already cited in 07.01.04 Master tier.
Burnside 1897/1904 — William Burnside, Theory of Groups of Finite Order (Cambridge 1897, 2nd ed. 1911); On groups of order $p^{α} q^{β}$ , Proc. London Math. Soc. (2) 1 (1904) 388–392. Burnside's $p^{a} q^{b}$ solvability theorem is a canonical early application of character theory. Cite in 07.01.03 Master tier (already present as forward reference).
Mackey 1949/1951/1952 — George W. Mackey, Imprimitivity for representations of locally compact groups, I, Proc. Natl. Acad. Sci. 35 (1949) 537–545; On induced representations of groups, Amer. J. Math. 73 (1951) 576–592; Induced representations of locally compact groups, I, Ann. Math. 55 (1952) 101–139, II, Ann. Math. 58 (1953) 193–221. Cite in new 07.01.09 mackey-… unit Master tier; already partially cited in 07.01.07.
Artin 1923/1930 — Emil Artin, Über eine neue Art von L-Reihen, Abh. Math. Sem. Univ. Hamburg 3 (1923) 89–108; Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren, Abh. Math. Sem. Univ. Hamburg 8 (1930) 292–306. Originates the Artin $L$ -function and Artin's induction theorem (1930). Cite in new 07.01.10 unit Master tier.
Brauer 1935/1946/1947 — Richard Brauer, Über die Darstellung von Gruppen in Galois'schen Feldern, Math. Ann. 110 (1935) 686–699 (modular rep theory); On Artin's L-series with general group characters, Ann. Math. 48 (1947) 502–514 (Brauer's induction theorem). The 1946 conference paper at the Princeton Bicentennial is the announcement; the 1947 Annals paper is the definitive publication. Cite in new 07.01.11 and 07.02.02/07.02.04 unit Master tiers. Already partially cited in 07.02.01 and 07.01.03.

Notation crosswalk. LRFG uses standard French-Bourbaki notation: $G$ for the group, $ρ : G \to GL (V)$ for the representation, $χ_{ρ}$ or $χ_{V}$ for the character, $\langle f, g \rangle_G = (1/|G|) \sum_x f(x) \overline{g(x)}$ for the inner product, $Ind_{H}^{G}$ and $Res_{H}^{G}$ for induction and restriction (some editions use $f_{*}$ and $f^{*}$ ). For modular reps: $(K, \mathcal{O}, k) $f or t h e$ p $- m o d u l a r t r i pl e;$ R_K(G), R_k(G), P_k(G)$ for the three Grothendieck groups; $β_{V}$ for the Brauer character of $V$ ; $c, d, e$ for the cde-triangle maps. Codex already uses Serre's ordinary notation (verified across 07.01.01–08); the new modular units 07.02.02–06 should adopt Serre's $(K, O, k)$ notation verbatim and adopt $R_{K} (G), R_{k} (G), P_{k} (G)$ with explicit definition in a §Notation paragraph of 07.02.02. The Brauer-character variable name $β$ (vs. $χ$ for ordinary characters) should be standardised in the new units to disambiguate.

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

07.01.07-induced-representation — 07.01.09–11 are direct follow-ups; rewrite the forward-reference commentary blocks at lines 412, 458, 463 in 07.01.07 to cite the new anchor units.
07.02.01-maschke-theorem — 07.02.02–06 close the forward-reference loop that the Master tier of Maschke opens. Rewrite the Master-tier blocks to cite anchor units rather than external Curtis-Reiner / Alperin / Navarro.
07.05.03-specht-module — modular Specht modules and the $S_{n}$ decomposition matrix become well-defined once 07.02.02–04 ship; rewrite the modular-rep forward references.
number-theory/ (if/when shipped) — 07.01.10 Artin induction
- 07.01.11 Brauer induction are the bridge to Artin $L$ -functions and the local-global principles of class field theory.
physics/quantum-mechanics/ (small-group symmetry) — 07.02.E1 worked character tables of $A_{4}, S_{4}, A_{5}$ are exactly the point-group character tables used in crystallography and molecular physics.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem in LRFG (full P1 audit; deferred until a local PDF is acquired). Coverage estimates in §2 are based on LRFG's well-known three-part chapter structure plus spot-checks of Codex units via the 9+ existing Serre LRFG citations.
Serre's two Lie-theoretic siblings (FT 3.12 Complex Semisimple Lie Algebras CSLA; FT 3.13 Lie Algebras and Lie Groups LALG). Both audited in Cycle 5; the plans plans/fasttrack/serre-complex-semisimple-lie-algebras.md and plans/fasttrack/serre-lie-algebras-and-lie-groups.md exist. The Serre slot triangle (3.12, 3.13, 3.15) is now fully audit-covered.
Curtis-Reiner Vols. I–II (canonical encyclopaedic reference). Deferred to a hypothetical future master-tier algebraic-groups / finite-group-theory audit. LRFG Parts I–III are a foundational gateway to Curtis-Reiner; closing the LRFG gap is the prerequisite for any Curtis-Reiner-scale audit.
Alperin Local Representation Theory (modern modular structure theory). Deferred. LRFG Part III is the prerequisite anchor; once 07.02.02–06 ship, an Alperin audit becomes feasible as the Master-tier deepening to modular block theory (Brauer pairs, Green correspondence, the source algebra formalism, defect theory).
Isaacs Character Theory of Finite Groups (depth-deepening of LRFG Parts I–II). Deferred. Isaacs covers Frobenius groups, $M$ -groups, $p$ -solvable groups, and the Brauer-Suzuki theorem in depth — material that LRFG mentions only in passing. An Isaacs audit is a natural follow-up to the LRFG Priority-1+2 punch-list.
Navarro Character Theory and the McKay Conjecture (2018). Modern monograph on the modular-rep / McKay-conjecture programme. Already referenced from 07.02.01 Master tier. Deferred to a research-track audit.
Diaconis Probability and Representation Theory (FT 3.16). Own audit; LRFG provides the finite-group foundation but Diaconis's random-walks-on-groups application is outside LRFG scope.
Modular representation theory of algebraic groups (Jantzen Representations of Algebraic Groups; Steinberg tensor product theorem; Frobenius kernels). LRFG covers only abstract finite groups; algebraic-group modular reps are a separate research area. Deferred to a hypothetical algebraic-groups/ chapter.
Hopf-algebraic and Tannakian formulations of finite-group reps (Drinfel'd double, fusion categories, modular tensor categories). Deferred to the planned 05-category-theory/ strand.
Reductive $p$ -adic group representation theory (Bernstein- Zelevinsky, the local Langlands programme for $GL_{n}$ ). LRFG provides the finite-group analogue but the $p$ -adic theory is its own research area. Deferred to a future Bushnell-Henniart / Bump audit.

§6 Acceptance criteria for FT equivalence (LRFG)

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

The 5 Priority-1 new units have shipped (07.02.02 modular-representation, 07.02.03 grothendieck-cde-triangle, 07.02.04 brauer-character, 07.01.10 artin-induction-theorem, 07.01.11 brauer-induction-theorem). These close the largest structural gaps.
≥95% of LRFG's named theorems and worked examples in chapters 1–18 map to a Codex unit (currently ~55%; after Priority-1 this rises to ~85%; after Priority-1+2 to ~92%; full ≥95% requires Priority-3 master deepenings plus the Priority-4 character-table pack).
≥90% of LRFG's worked computations have a direct unit or are cross-referenced from a unit that covers them. Specifically: the small-group character tables ( $A_{4}, S_{4}, A_{5}, Q_{8}$ , Heisenberg) should be in 07.02.E1; the $G L_{2} (F_{q})$ rep-theory example (LRFG §8) should be in 07.01.09 or 07.01.10; the $S_{n}$ rep-theory via Young subgroups (LRFG §8) is partially in 07.05.01–03 but the induction-from-Young-subgroups angle should be cross-referenced.
The silent-dependency forward-references in 07.01.01, 07.01.03, 07.01.04, 07.01.07, 07.01.08, 07.02.01, 07.05.03 are rewritten to cite the new anchor units (Priority 1 ships ⇒ this is automatic, but the rewrite is a separate Pass-W task). This is the only acceptance criterion that requires editing existing shipped units, not just adding new ones.
Notation crosswalk recorded (see §4) in 07.02.02 §Notation paragraph.
Originator-prose sections (Frobenius 1896, Schur 1905, Mackey 1949–52, Artin 1923/1930, Brauer 1935/1946) appear in the relevant new-unit Master tiers; existing citations in 07.01.03, 07.01.04, 07.02.01 Master tiers should be consolidated and cross-referenced.

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

4.1 Book-as-input self-check. Sample 5 random theorem statements from LRFG; for each, identify the Codex unit that proves it. Particularly stress-test the Part III sample (modular section, which is the largest gap).
4.2 Exercise reproducibility. Sample 5 random exercises from LRFG. LRFG exercises are short and computational (unlike Fulton- Harris); the sampling should bias toward Ch. 2 (orthogonality), Ch. 7 (Mackey decomposition), and Ch. 15 (decomposition matrices) to spot-check the punch-list units.
4.3 Notation comprehension. Sample 5 fragments of LRFG prose (one per part: I, II, III; plus two boundary fragments); paraphrase each into the Codex notation system (per §4 crosswalk).

§7 Sourcing

Local PDF status. Not present in reference/textbooks-extra/ (verified — directory listing confirms no Serre LRFG file; only the cover image reference/fast-track/images/Serre-Lie-Algebras-Lie-Groups-712x1024__dd9fad6ce3.jpg is local, and that cover image is shared across multiple Serre Fast Track slots and corresponds to Lie Algebras and Lie Groups, not LRFG). LRFG is a commercial Springer GTM (GTM 42, 1977 English translation by Leonard L. Scott of Représentations linéaires des groupes finis, Hermann 1967/2nd ed. 1971) and is not author-hosted.
Commercial source. Springer GTM 42. Print + Springer eBook available at link.springer.com/book/10.1007/978-1-4684-9458-7. ISBN 0-387-90190-6 (hardcover), 978-1-4684-9458-7 (eBook).
Library-mirror sources. Acquire via institutional access (university library Springer subscription) or interlibrary loan. For the local copy, target placement is reference/textbooks-extra/Serre-LinearRepresentationsFiniteGroups.pdf to mirror the pattern of other commercial-source FT texts.
Anna's Archive. A search at annas-archive.org/search?q=serre+linear+representations+finite+groups typically yields multiple OCR'd copies of GTM 42; acquisition is deferred to a follow-up sourcing pass (not done in this audit per the 3-hour time limit).
French original. Serre, Représentations linéaires des groupes finis, Hermann (Collection Méthodes), Paris 1967; 2nd ed. 1971; 3rd ed. (revised) 1978. Bibliothèque Nationale de France hosts a digital copy of the 1st edition in restricted-access form. The 1977 English translation is the standard reference internationally.
Companion / supplementary materials.
- I. M. Isaacs, Character Theory of Finite Groups (Academic Press 1976; AMS Chelsea reprint 2006). The depth-deepening of LRFG Parts I–II.
- C. W. Curtis, I. Reiner, Methods of Representation Theory Vols. I (1981) and II (1987). Encyclopaedic.
- J. L. Alperin, Local Representation Theory (Cambridge Studies in Advanced Mathematics 11, 1986). Modern modular treatment.
- G. James, M. Liebeck, Representations and Characters of Groups (Cambridge, 2nd ed. 2001). Gentle undergraduate exposition of Part I.
- G. Navarro, Character Theory and the McKay Conjecture (Cambridge Studies in Advanced Mathematics 175, 2018). Modern research-level sequel.
Open-access alternatives covering ~70% of LRFG:
- P. Etingof et al., Introduction to Representation Theory (AMS Student Mathematical Library 59, 2011; free PDF at arxiv.org/abs/0901.0827). Covers LRFG Parts I–II at gentler pace with cleaner category-theoretic spine; does not cover modular rep theory.
- K. Conrad, Notes on Representation Theory (Yale lecture notes, free, hosted at kconrad.math.uconn.edu/blurbs/). Multiple notes covering Mackey theory, Brauer induction, and Artin induction at the LRFG Part-II level.
- MIT OCW 18.715 (Pavel Etingof, Introduction to Representation Theory, free). Covers LRFG Parts I–II; modular section absent.
Reduced-audit flag. This plan was produced without a local LRFG PDF. Coverage estimates in §2 should be re-verified once a PDF is acquired, particularly for the Part III chapters (LRFG §§14–18) where structural detail (the cde-triangle map definitions, the projective cover construction, the explicit statement and proof of Brauer's main theorems) is the load-bearing pedagogical content. Promote to full P1 audit when PDF is local.

Serre — *Linear Representations of Finite Groups* (Fast Track 3.15) — Audit + Gap Plan