{ "schema": "rt-fin-source-dispositions/v1", "edition": "2026-10-05", "scope": "Exact source identities, compared reference scope and proof-provider states for the17-lesson edition. Source availability does not certify a complete reading or recursive foundational proof closure.", "lessons": [ { "lesson": "RT-FIN-01", "title": "Representations and complete reducibility", "source": "src/RT-FIN-01.md", "source_sha256": "E27AD6CC52BDA47B09D9DC9CAE4C4D072E701600FBC4B9B50817F72E1A152BFC", "proof_scope": "Maschke, invariant Hermitian forms, Schur and matrix modules", "disposition": "Elementary inputs are proved in Section 0. The Maschke, Schur and invariant-inner-product arguments were compared with free sources; original theorem scope and all five solved exercises are retained. This is not whole-course or recursive foundation closure.", "reference_leads": [ { "authors": "Peter Webb", "title": "A Course in Finite Group Representation Theory", "version": "2016; author’s prepublication draft, 2016-02-23", "url": "https://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf", "access": "free to read", "covers": "§1.1, Definitions and Examples; §1.2, Semisimple Representations.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "Pavel Etingof; Oleg Golberg; Sebastian Hensel; Tiankai Liu; Alex Schwendner; Dmitry Vaintrob; Elena Yudovina", "title": "Introduction to Representation Theory", "version": "2010; MIT 18.712, Fall 2010 lecture notes", "url": "https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/pages/lecture-notes/", "access": "open licence: CC BY-NC-SA 4.0", "covers": "§1.3, representations; §3.1, Maschke’s theorem.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "comparisons_read": [ { "url": "https://arxiv.org/pdf/0901.0827", "scope": "Proposition 1.16 and Corollary 1.17; Maschke in Section 3.1; Theorems 3.11 and 3.12.", "use": "Independent proof comparison, no copied wording." }, { "url": "https://math.berkeley.edu/~serganov/math252/Bookrep.pdf", "scope": "Seven-chapter draft, Section 3, Maschke and complete reducibility.", "use": "Independent proof comparison; its Lemma 3.10 complement shortcut is not used." } ] }, { "lesson": "RT-FIN-02", "title": "Characters and the orthogonality relations", "source": "src/RT-FIN-02.md", "source_sha256": "1078B88109DD46DBA17949F19A228C5825FBD3DB21F463426BFD43B25E602EA0", "proof_scope": "Matrix and character orthogonality, regular multiplicities and the class-function basis", "disposition": "All existing character results, five tables and five worked solutions retained. Exact internal prerequisites and complete elementary proofs supplied; character arguments independently checked against the free Etingof notes. Recursive starting foundations remain explicitly separate.", "reference_leads": [ { "authors": "Peter Webb", "title": "A Course in Finite Group Representation Theory", "version": "2016; author’s prepublication draft, 2016-02-23", "url": "https://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf", "access": "free to read", "covers": "§§3.1–3.4: character tables, orthogonality relations and the number of irreducible characters.", "disposition": "Compared complete row-orthogonality and multiplicity proofs at printed pp. 28–31 (physical pp. 35–38); retain the complete independently authored proof." }, { "authors": "Pavel Etingof; Oleg Golberg; Sebastian Hensel; Tiankai Liu; Alex Schwendner; Dmitry Vaintrob; Elena Yudovina", "title": "Introduction to Representation Theory", "version": "2010; MIT 18.712, Fall 2010 lecture notes", "url": "https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/pages/lecture-notes/", "access": "open licence: CC BY-NC-SA 4.0", "covers": "§3.5, character orthogonality; §3.7, matrix-element orthogonality.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "comparisons_read": [ { "url": "https://arxiv.org/pdf/0901.0827", "scope": "Printed37–40, complete orthogonality/matrix-coefficient arguments; rendered pages37,38,40 visually checked. Earlier3.1/3.4 read. All current lesson arguments checked independently.", "use": "No source wording copied." }, { "url": "https://math.berkeley.edu/~serganov/math252/Bookrep.pdf", "scope": "Author draft title and Chapter1 Section4; comparison/further reading only.", "use": "No source wording copied." }, { "url": "https://www.jmilne.org/math/CourseNotes/gt.html", "scope": "GT400 native source ga07/ga08: complete coset-to-orbit proof.", "use": "Independent exposition, not a imported source passage." } ] }, { "lesson": "RT-FIN-03", "title": "The group algebra and Fourier analysis on a finite group", "source": "src/RT-FIN-03.md", "source_sha256": "548DD8D92C19E4F59F55AE2C9DBFD933D7F84852382F8D2AD2DF1BE2B17844D3", "proof_scope": "Fourier/Wedderburn isomorphism, central idempotents and convolution", "disposition": "Added full elementary coset/orbit/quotient proofs, a direct A5 six-subgroup count, the fifth-root minimal-polynomial proof and finite-order spectral projections. Fourier blocks and idempotents are proved directly; retired the unverified external prerequisite routes.", "reference_leads": [ { "authors": "Peter Webb", "title": "A Course in Finite Group Representation Theory", "version": "2016; author’s prepublication draft, 2016-02-23", "url": "https://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf", "access": "free to read", "covers": "§3.6, The Matrix Summands of the Complex Group Algebra.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "Evan DeCorte; David de Laat; Frank Vallentin", "title": "Fourier analysis on finite groups and the Lovász theta-number of Cayley graphs", "version": "2013; arXiv v1, 2013-07-22", "url": "https://arxiv.org/abs/1307.5703", "access": "free to read", "covers": "§2, Definitions, notation, and background in Fourier analysis: Fourier transforms on arbitrary finite groups.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "mailbox_candidate_resolution": { "flag": "check_candidate", "status": "resolved_by_owner", "observed_utc": "2026-10-05T02:37:12.870326+00:00", "scope": "Added full elementary coset/orbit/quotient proofs, a direct A5 six-subgroup count, the fifth-root minimal-polynomial proof and finite-order spectral projections. Fourier blocks and idempotents are proved directly; retired the unverified external prerequisite routes.", "new_unadmitted_body_reading": false } }, { "lesson": "RT-FIN-04", "title": "Tensor products, duals and real representations", "source": "src/RT-FIN-04.md", "source_sha256": "727A564AC7F811AF120FBDD3E6AB0DD75C0FF228769CE33C375A81D20987F7F6", "proof_scope": "Tensor/dual characters and real, complex and quaternionic types", "disposition": "Retained the complete independently authored and self-checked lesson; no warranted replacement or new adapted expression.", "reference_leads": [ { "authors": "Pavel Etingof; Oleg Golberg; Sebastian Hensel; Tiankai Liu; Alex Schwendner; Dmitry Vaintrob; Elena Yudovina", "title": "Introduction to Representation Theory", "version": "2010; MIT 18.712, Fall 2010 lecture notes", "url": "https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/pages/lecture-notes/", "access": "open licence: CC BY-NC-SA 4.0", "covers": "§3.4, duals and tensor products; §4.1, Frobenius–Schur indicator and real/quaternionic types.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "Constantin Teleman", "title": "Representation Theory", "version": "2005; Lent 2005 lecture notes", "url": "https://math.berkeley.edu/~teleman/math/RepThry.pdf", "access": "free to read", "covers": "§8, especially §§8.6–8.8: representation rings, tensor products, duals and complex-conjugate characters.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ] }, { "lesson": "RT-FIN-05", "title": "Integrality of characters and Burnside's \\(p^a q^b\\) theorem", "source": "src/RT-FIN-05.md", "source_sha256": "5F590186B2E2B9AB7C65558F82E2968C4E7F1628BC5500D8F38175C9C72D24D0", "proof_scope": "Character integrality, degree divisibility, normalized traces and Burnside solvability", "disposition": "Retained complete fixed-field, algebraic-integer, Sylow and Burnside proofs. Added the integer-matrix Cayley–Hamilton proof and linked group counting to Lesson 3 Lemma 6.1.", "reference_leads": [ { "authors": "Peter Webb", "title": "A Course in Finite Group Representation Theory", "version": "2016; author’s prepublication draft, 2016-02-23", "url": "https://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf", "access": "free to read", "covers": "§3.5, algebraic integers and character-degree divisibility; §3.7, Burnside’s theorem.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "Constantin Teleman", "title": "Representation Theory", "version": "2005; Lent 2005 lecture notes", "url": "https://math.berkeley.edu/~teleman/math/RepThry.pdf", "access": "free to read", "covers": "§18: algebraic-integrality arguments and Burnside’s theorem for groups of order p^a q^b.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "mailbox_candidate_resolution": { "flag": "check_candidate", "status": "resolved_by_owner", "observed_utc": "2026-10-05T02:37:12.870326+00:00", "scope": "Retained complete fixed-field, algebraic-integer, Sylow and Burnside proofs. Added the integer-matrix Cayley–Hamilton proof and linked group counting to Lesson 3 Lemma 6.1.", "new_unadmitted_body_reading": false } }, { "lesson": "RT-FIN-06", "title": "Induced representations and Frobenius reciprocity", "source": "src/RT-FIN-06.md", "source_sha256": "98475FE12E0C38592A1847D7FD308F0F60AE336F3A5C9F1B274BF360572B7EFD", "proof_scope": "Function and tensor induction, both adjunctions, transitivity and projection", "disposition": "Retained both fully explicit Frobenius adjunctions and replaced the coset-count import by the complete Lesson 3 Lemma 6.1 proof.", "reference_leads": [ { "authors": "Peter Webb", "title": "A Course in Finite Group Representation Theory", "version": "2016; author’s prepublication draft, 2016-02-23", "url": "https://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf", "access": "free to read", "covers": "§4.3, Induction and Restriction, including Frobenius reciprocity.", "disposition": "Compared both adjunctions, transitivity and projection at printed pp. 58–60 (physical pp. 65–67); retain the more explicit course verification of maps." }, { "authors": "Constantin Teleman", "title": "Representation Theory", "version": "2005; Lent 2005 lecture notes", "url": "https://math.berkeley.edu/~teleman/math/RepThry.pdf", "access": "free to read", "covers": "§14, induced representations; §15, induced characters and Frobenius reciprocity.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "mailbox_candidate_resolution": { "flag": "check_candidate", "status": "resolved_by_owner", "observed_utc": "2026-10-05T02:37:12.870326+00:00", "scope": "Retained both fully explicit Frobenius adjunctions and replaced the coset-count import by the complete Lesson 3 Lemma 6.1 proof.", "new_unadmitted_body_reading": false } }, { "lesson": "RT-FIN-07", "title": "Mackey theory and Clifford's theorem", "source": "src/RT-FIN-07.md", "source_sha256": "A7D857075598F96B6258796133C7C13021D23D6A5DCA41915428315AA38567C9", "proof_scope": "Mackey double cosets, irreducibility and the full inertia correspondence", "disposition": "Retained complete Mackey/Clifford proofs and replaced both coset and subgroup-orbit imports by Lesson 3 Lemma 6.1.", "reference_leads": [ { "authors": "Peter Webb", "title": "A Course in Finite Group Representation Theory", "version": "2016; author’s prepublication draft, 2016-02-23", "url": "https://www-users.math.umn.edu/~webb/RepBook/RepBookLatex.pdf", "access": "free to read", "covers": "§5.1, double cosets; §5.2, Mackey’s theorem; §5.3, Clifford’s theorem.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "Constantin Teleman", "title": "Representation Theory", "version": "2005; Lent 2005 lecture notes", "url": "https://math.berkeley.edu/~teleman/math/RepThry.pdf", "access": "free to read", "covers": "§16, Mackey theory and the irreducibility criterion for induced representations.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "mailbox_candidate_resolution": { "flag": "check_candidate", "status": "resolved_by_owner", "observed_utc": "2026-10-05T02:37:12.870326+00:00", "scope": "Retained complete Mackey/Clifford proofs and replaced both coset and subgroup-orbit imports by Lesson 3 Lemma 6.1.", "new_unadmitted_body_reading": false } }, { "lesson": "RT-FIN-08", "title": "Groups with an abelian normal subgroup: the little-group method", "source": "src/RT-FIN-08.md", "source_sha256": "6165D86B5579B2DEBD22F9D69F9EFD4EA91839B20DE56B6004C46A659B27CCA6", "proof_scope": "Abelian-normal weights, little groups and the finite Heisenberg representations", "disposition": "The finite split little-group and Heisenberg classifications are proved in the lesson. Teleman supplies the checked normal-abelian antecedent and Gruson–Serganova supplies the cited comparison scope.", "reference_leads": [ { "authors": "Amritanshu Prasad; M. K. Vemuri", "title": "Mackey’s Little Group Method", "version": "2007; notes dated 2007-06-29", "url": "https://ncatlab.org/nlab/files/PrasadVemuri-MackeyMethod.pdf", "access": "free to read", "covers": "§§2–3, systems of imprimitivity and Mackey’s theorem; §4, Application to Representation Theory, especially Proposition 4.2.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "Pavel Etingof; Oleg Golberg; Sebastian Hensel; Tiankai Liu; Alex Schwendner; Dmitry Vaintrob; Elena Yudovina", "title": "Introduction to Representation Theory", "version": "2010; MIT 18.712, Fall 2010 lecture notes", "url": "https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/pages/lecture-notes/", "access": "open licence: CC BY-NC-SA 4.0", "covers": "§4.26, Semidirect Products: the little-group construction with abelian normal subgroup.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "mailbox_candidate_resolution": { "flag": "check_candidate", "status": "resolved_by_owner", "observed_utc": "2026-10-05T02:37:12.870326+00:00", "scope": "The finite split little-group and Heisenberg classifications are proved in the lesson. Teleman supplies the checked normal-abelian antecedent and Gruson–Serganova supplies the cited comparison scope.", "new_unadmitted_body_reading": false } }, { "lesson": "RT-FIN-09", "title": "Frobenius groups and Frobenius's theorem", "source": "src/RT-FIN-09.md", "source_sha256": "6AE8D3AFA42C5B03174AD94725B899A575F8650383681D252B66989F6FD051D1", "proof_scope": "Frobenius kernels and complements", "disposition": "Retained the complete independently authored and self-checked lesson; no warranted replacement or new adapted expression.", "reference_leads": [ { "authors": "Hendrik W. Lenstra; Jeanine Daems; Willem Jan Palenstijn", "title": "Representatietheorie", "version": "2003; course notes", "url": "https://websites.math.leidenuniv.nl/algebra/rpt.pdf", "access": "free to read", "covers": "§12, Frobenius’ theorem: the character-theoretic construction of the Frobenius kernel.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "Terence Tao", "title": "The theorems of Frobenius and Suzuki on finite groups", "version": "2013; exposition dated 2013-04-12", "url": "https://terrytao.wordpress.com/2013/04/12/the-theorems-of-frobenius-and-suzuki-on-finite-groups/", "access": "free to read", "covers": "Theorem 2 and its proof: Frobenius’s theorem; Theorem 3 gives the permutation-group formulation.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ] }, { "lesson": "RT-FIN-10", "title": "Artin's induction theorem and rationality", "source": "src/RT-FIN-10.md", "source_sha256": "4F4F4F861251BB48DCF3273F9FF8A51627F7CEF1A85E96509D53CF64D4F36EC3", "proof_scope": "Cyclotomic irreducibility and rational Artin induction", "disposition": "The cyclotomic and rational Artin arguments are proved in the lesson. The exact2018 Gruson–Serganova Chapter2 §12 comparison is checked.", "reference_leads": [ { "authors": "Pavel Etingof; Oleg Golberg; Sebastian Hensel; Tiankai Liu; Alex Schwendner; Dmitry Vaintrob; Elena Yudovina", "title": "Introduction to Representation Theory", "version": "2010; MIT 18.712, Fall 2010 lecture notes", "url": "https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/pages/lecture-notes/", "access": "open licence: CC BY-NC-SA 4.0", "covers": "§4.25, Artin’s induction theorem.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "Kiyoshi Igusa", "title": "Math 101b: Representation Theory — Artin’s Induction Theorem", "version": "2007; Spring 2007 lecture notes", "url": "https://people.brandeis.edu/~igusa/Math101bS07/Math101b_notesD3c.pdf", "access": "free to read", "covers": "§3c, especially Theorem 3.18: Artin induction and rational linear combinations of induced characters.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "mailbox_candidate_resolution": { "flag": "check_candidate", "status": "resolved_by_owner", "observed_utc": "2026-10-05T02:37:12.870326+00:00", "scope": "The cyclotomic and rational Artin arguments are proved in the lesson. The exact2018 Gruson–Serganova Chapter2 §12 comparison is checked.", "new_unadmitted_body_reading": false } }, { "lesson": "RT-FIN-11", "title": "Brauer's induction theorem", "source": "src/RT-FIN-11.md", "source_sha256": "2D724B0D19A38E732781DA61DBC37ABB2D2344330DCBDF3610483CFCB125E7BF", "proof_scope": "Nilpotent monomiality, local induction and full integer Brauer induction", "disposition": "The full integer linear-character Brauer theorem is proved in the lesson. The exact free Kramar §3 comparison is checked, including p-regular congruence and induction normalization.", "reference_leads": [ { "authors": "Minatsu Yamaji", "title": "Applications of Brauer Induction to Artin L-Functions", "version": "1996; MSc thesis, McGill University", "url": "https://escholarship.mcgill.ca/downloads/fj236447x", "access": "free to read", "covers": "§2.2, Brauer’s Induction Theorem, beginning on p. 15.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "Robert Boltje", "title": "A canonical Brauer induction formula", "version": "1990; Astérisque 181–182, 31–59", "url": "https://www.numdam.org/article/AST_1990__181-182__31_0.pdf", "access": "free to read", "covers": "Complete paper, pp. 31–59: a canonical, explicit refinement of Brauer induction.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "mailbox_candidate_resolution": { "flag": "check_candidate", "status": "resolved_by_owner", "observed_utc": "2026-10-05T02:37:12.870326+00:00", "scope": "The full integer linear-character Brauer theorem is proved in the lesson. The exact free Kramar §3 comparison is checked, including p-regular congruence and induction normalization.", "new_unadmitted_body_reading": false } }, { "lesson": "RT-FIN-12", "title": "Consequences of Brauer's theorem: characterization of characters and splitting fields", "source": "src/RT-FIN-12.md", "source_sha256": "54CB3881B34AE3243E82537BA7BFB7BE54A7D60068DF136AB011350B4F49272F", "proof_scope": "Character criteria and splitting-field realization", "disposition": "Retained and directly audited character recognition and norm-one splitting-field extraction from integer Brauer induction. Replaced external arithmetic imports by exact NT-CFT 21 results; the NT-ADL-10 analytic text is recorded as owner-supplied but absent from this reader, so the arithmetic continuation aside remains conditional.", "reference_leads": [ { "authors": "Hendrik W. Lenstra; Jeanine Daems; Willem Jan Palenstijn", "title": "Representatietheorie", "version": "2003; course notes", "url": "https://websites.math.leidenuniv.nl/algebra/rpt.pdf", "access": "free to read", "covers": "§14, Inductie en de stelling van Brauer; Corollary 14.5 gives the cyclotomic splitting-field consequence.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "Frieder Ladisch", "title": "Answer to “Characterization of group characters”", "version": "2017; MathOverflow answer, revised 2017-03-14 at 15:55", "url": "https://mathoverflow.net/questions/264554/characterization-of-group-characters", "access": "free to read", "covers": "Complete answer: Brauer’s elementary-subgroup criterion for generalized characters and the additional irreducibility test.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "mailbox_candidate_resolution": { "flag": "check_candidate", "status": "resolved_by_owner", "observed_utc": "2026-10-05T02:37:12.870326+00:00", "scope": "Retained and directly audited character recognition and norm-one splitting-field extraction from integer Brauer induction. Replaced external arithmetic imports by exact NT-CFT 21 results; the NT-ADL-10 analytic text is recorded as owner-supplied but absent from this reader, so the arithmetic continuation aside remains conditional.", "new_unadmitted_body_reading": false } }, { "lesson": "RT-FIN-13", "title": "The symmetric groups I: Young tableaux and Young symmetrizers", "source": "src/RT-FIN-13.md", "source_sha256": "3BF691EFEE4CBE71BB600039BF9896366F954438A0DC0B295298483F1517EC2B", "proof_scope": "Young symmetrizers, irreducibles and standard-tableau bases", "disposition": "Rechecked row/column collision, fixed-tableau factorization, coefficient symmetry and nonzero trace normalization against admitted 2018 Gruson–Serganova Chapter 6 §1. Preserved every complete proof and rational model; stated the exact product-order and normalization obligations.", "reference_leads": [ { "authors": "Pavel Etingof; Oleg Golberg; Sebastian Hensel; Tiankai Liu; Alex Schwendner; Dmitry Vaintrob; Elena Yudovina", "title": "Introduction to Representation Theory", "version": "2010; MIT 18.712, Fall 2010 lecture notes", "url": "https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/pages/lecture-notes/", "access": "open licence: CC BY-NC-SA 4.0", "covers": "§§4.12–4.13, symmetric-group representations and the Young-symmetrizer construction.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "T. Geetha; Amritanshu Prasad", "title": "Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups", "version": "2017; arXiv v2, 2017-05-22", "url": "https://arxiv.org/abs/1606.04424", "access": "free to read", "covers": "§2, Gelfand–Tsetlin bases: partitions, Young diagrams and standard Young tableaux; this source covers the tableau component.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "mailbox_candidate_resolution": { "flag": "rewrite_candidate", "status": "resolved_by_owner", "observed_utc": "2026-10-05T02:37:12.870326+00:00", "scope": "Rechecked row/column collision, fixed-tableau factorization, coefficient symmetry and nonzero trace normalization against admitted 2018 Gruson–Serganova Chapter 6 §1. Preserved every complete proof and rational model; stated the exact product-order and normalization obligations.", "new_unadmitted_body_reading": false } }, { "lesson": "RT-FIN-14", "title": "The symmetric groups II: branching, Jucys–Murphy elements and Young's seminormal form", "source": "src/RT-FIN-14.md", "source_sha256": "F921D67A9A1D8608CD65716E7FC83925FE36AA2F58EBC236A55CB265C8AAE8BE", "proof_scope": "Branching, seminormal form, Jucys–Murphy elements and hook dimensions", "disposition": "Rechecked the free Vershik–Okounkov v3 Theorem 5.8 and Propositions 6.1–6.2. Retained direct Coxeter, projector, branching and hook-recursion proofs; internal projector and hook proof routes now explicit.", "reference_leads": [ { "authors": "Anatoly M. Vershik; Andrei Yu. Okounkov", "title": "A New Approach to the Representation Theory of the Symmetric Groups. 2", "version": "2005; arXiv v3, 2005-04-20", "url": "https://arxiv.org/abs/math/0503040", "access": "free to read", "covers": "§§2–6: branching, Gelfand–Tsetlin algebras and Jucys–Murphy elements; §6, Propositions 6.1–6.2, seminormal and orthogonal forms.", "disposition": "Existing bounded direct comparison, including Theorem 5.8 and Propositions 6.1–6.2, remains applicable to the unchanged source; no new expression reused." }, { "authors": "T. Geetha; Amritanshu Prasad", "title": "Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups", "version": "2017; arXiv v2, 2017-05-22", "url": "https://arxiv.org/abs/1606.04424", "access": "free to read", "covers": "§2, Gelfand–Tsetlin bases: multiplicity-free branching and Young’s orthogonal form, especially equations (2)–(4).", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "mailbox_candidate_resolution": { "flag": "check_candidate", "status": "resolved_by_owner", "observed_utc": "2026-10-05T02:37:12.870326+00:00", "scope": "Rechecked the free Vershik–Okounkov v3 Theorem 5.8 and Propositions 6.1–6.2. Retained direct Coxeter, projector, branching and hook-recursion proofs; internal projector and hook proof routes now explicit.", "new_unadmitted_body_reading": false } }, { "lesson": "RT-FIN-15", "title": "The symmetric groups III: characters and symmetric functions", "source": "src/RT-FIN-15.md", "source_sha256": "4D3C0492505F7E25767AAB35DC59F7ADCAD4103F9C41418DB7820D39070E767E", "proof_scope": "Symmetric-function bases, Frobenius characteristic, Murnaghan–Nakayama and Littlewood–Richardson", "disposition": "Rechecked exact licensed Grinberg–Reiner Proposition 2.5.15, Corollary 2.5.17 and alternant/tableau results, and Stembridge’s column involution. Preserved the bounded CC BY kernel component and all independent full proofs. Corrected the old corollary number; showed finite-alphabet infinite H directly.", "reference_leads": [ { "authors": "Hanspeter Kraft; Claudio Procesi", "title": "Classical Invariant Theory: A Primer", "version": "1996; preliminary version, July 1996", "url": "https://dmi.unibas.ch/fileadmin/user_upload/dmi/Personen/Kraft_Hanspeter/Classical_Invariant_Theory.pdf", "access": "free to read", "covers": "§6.2, Schur polynomials; §6.5, characters of symmetric groups.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "Constantin Teleman", "title": "Representation Theory", "version": "2005; Lent 2005 lecture notes", "url": "https://math.berkeley.edu/~teleman/math/RepThry.pdf", "access": "free to read", "covers": "§24, Frobenius character formula: symmetric-group characters and symmetric functions.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "mailbox_candidate_resolution": { "flag": "rewrite_candidate", "status": "resolved_by_owner", "observed_utc": "2026-10-05T02:37:12.870326+00:00", "scope": "Rechecked exact licensed Grinberg–Reiner Proposition 2.5.15, Corollary 2.5.17 and alternant/tableau results, and Stembridge’s column involution. Preserved the bounded CC BY kernel component and all independent full proofs. Corrected the old corollary number; showed finite-alphabet infinite H directly.", "new_unadmitted_body_reading": false } }, { "lesson": "RT-FIN-16", "title": "Schur–Weyl duality", "source": "src/RT-FIN-16.md", "source_sha256": "B272EC20CD6B5C236BCFC5C68EE9D078404940CA9E82A4AD87446E5A044B6713", "proof_scope": "Double centralizers, Schur–Weyl duality and polynomial general-linear representations", "disposition": "Rechecked 2018 Gruson–Serganova Chapter 6 §2. Preserved full finite density, double centralizers, tensor nonvanishing and polynomial classification. Character/dimension proofs now identify exact earlier symmetric-function tools and distinguish polynomial from rational conventions.", "reference_leads": [ { "authors": "Hanspeter Kraft; Claudio Procesi", "title": "Classical Invariant Theory: A Primer", "version": "1996; preliminary version, July 1996", "url": "https://dmi.unibas.ch/fileadmin/user_upload/dmi/Personen/Kraft_Hanspeter/Classical_Invariant_Theory.pdf", "access": "free to read", "covers": "§§3.1–3.3, centralizers, double-centralizer theorem and tensor-space decomposition.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." }, { "authors": "Pavel Etingof; Oleg Golberg; Sebastian Hensel; Tiankai Liu; Alex Schwendner; Dmitry Vaintrob; Elena Yudovina", "title": "Introduction to Representation Theory", "version": "2010; MIT 18.712, Fall 2010 lecture notes", "url": "https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/pages/lecture-notes/", "access": "open licence: CC BY-NC-SA 4.0", "covers": "§§4.18–4.19, Schur–Weyl duality and its GL(V) formulation.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ], "mailbox_candidate_resolution": { "flag": "rewrite_candidate", "status": "resolved_by_owner", "observed_utc": "2026-10-05T02:37:12.870326+00:00", "scope": "Rechecked 2018 Gruson–Serganova Chapter 6 §2. Preserved full finite density, double centralizers, tensor nonvanishing and polynomial classification. Character/dimension proofs now identify exact earlier symmetric-function tools and distinguish polynomial from rational conventions.", "new_unadmitted_body_reading": false } }, { "lesson": "RT-FIN-17", "title": "Representations of GL₂ over finite fields", "source": "src/RT-FIN-17.md", "source_sha256": "E0A90A19BA1AD2E71DB2635586B0F0E48070C69E6639F9A50330D8B9EA08964C", "proof_scope": "All-prime-power GL2 classes, principal series, Steinberg and cuspidal characters", "disposition": "Retained the complete independently authored and self-checked lesson; no warranted replacement or new adapted expression.", "reference_leads": [ { "authors": "Paul Garrett", "title": "Representations of GL(2) and SL(2) over finite fields", "version": "2009; notes dated 2009-04-19", "url": "https://www.math.umn.edu/~garrett/m/v/toy_GL2.pdf", "access": "free to read", "covers": "§§2–5: GL₂ principal series, Whittaker models, uniqueness and the GL₂ summary.", "disposition": "A challenge/interstitial is not proof access. Not used to certify or reconstruct this lesson." }, { "authors": "Pavel Etingof; Oleg Golberg; Sebastian Hensel; Tiankai Liu; Alex Schwendner; Dmitry Vaintrob; Elena Yudovina", "title": "Introduction to Representation Theory", "version": "2010; MIT 18.712, Fall 2010 lecture notes", "url": "https://ocw.mit.edu/courses/18-712-introduction-to-representation-theory-fall-2010/pages/lecture-notes/", "access": "open licence: CC BY-NC-SA 4.0", "covers": "§4.24, representations of GL₂ over a finite field.", "disposition": "Reference lead retained; no expression adapted and no newly completed reading claimed for this lead." } ] } ] }