# Reading roadmap: Quadratic stabilization of the canonical Foulkes--Howe map The manuscript expresses the specified canonical map as multiplication in an invariant tensor algebra. An annihilator criterion reduces surjectivity to vanishing of symmetric multilinear forms on products. A directional-derivative injectivity lemma and two differentiation sequences surround an induction on the number of slots, yielding the stated degree range. Complete reducibility then supplies a representation embedding in the opposite direction. Source SHA-256: `89b46306d4bc756ff5fc261deaf3075ce1afa37087a313bab7abaf1569b8af21`. Indexed 24,462 characters. Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates. ## Specify the canonical image Use averaging and multiplication to define the map and its image, then express surjectivity through an annihilator. Claims: thm:main, sec:map ## Establish one-step injectivity Use adjoint directional derivatives and a norm identity on homogeneous spaces. Claims: lem:shift ## Run the slot induction Combine binary splitting, two derivative sequences and the induction hypothesis to obtain multilinear vanishing. Claims: prop:vanishing, thm:main ## Derive the representation comparison Split the surjection equivariantly and state the scope of the construction. Claims: cor:comparison, remark:line:462 ## Internal claim index ### Theorem 1 — `thm:main` Source lines 99–104. Apply the local multilinear vanishing proposition to the annihilator criterion for the canonical image, giving the stated surjectivity range. - `prop:vanishing` (proof_dependency; source_audit_annotation): The theorem proof invokes the local vanishing proposition. - `sec:map` (proof_dependency; source_audit_annotation): The theorem proof applies the canonical-image annihilator criterion. Candidate references: `eq:map` (internal_reference); `prop:vanishing` (candidate_proof_dependency); `sec:map` (candidate_proof_dependency). ### Lemma 2 — `lem:shift` Source lines 260–266. Make opposite directional derivatives adjoints under the monomial factorial inner product. Their commutator gives the degree difference, so the norm identity forces a zero kernel when the source degree exceeds the destination degree. Candidate references: `external:AldazRender` (external_reference). ### Proposition 3 — `prop:vanishing` Source lines 304–314. Induct on the number of slots. Binary splitting yields a common-product identity; a first derivative sequence frees the last slot, induction separates the other products, and a second sequence removes their shared power within the degree bound. - `lem:shift` (proof_dependency; source_audit_annotation): The first sequence checks source and destination degrees and applies injectivity at each step. - `sec:map` (proof_dependency; source_audit_annotation): The final step uses the earlier polarization proof that pure powers span the symmetric power. Candidate references: `eq:diagonal` (candidate_proof_dependency); `eq:binary` (candidate_proof_dependency); `lem:shift` (candidate_proof_dependency); `eq:inductive-vanishing` (candidate_proof_dependency); `eq:second-composition` (candidate_proof_dependency); `sec:map` (candidate_proof_dependency). ### Corollary 4 — `cor:comparison` Source lines 440–449. Use the main theorem’s equivariant surjection and complete reducibility to choose an invariant complement to its kernel; invert the restriction to construct the claimed embedding. - `thm:main` (proof_dependency; source_audit_annotation): The corollary uses the main theorem’s surjective map and its equivariance. - `external:Macdonald` (proof_dependency; source_audit_annotation): The cited complete-reducibility result supplies the invariant complement used to split the surjection. Candidate references: `thm:main` (candidate_proof_dependency); `external:Macdonald` (external_reference). ### Remark 5 — `remark:line:462` Source lines 462–472. State that the degree comparisons are independent of dimension and list the written proof’s use of the Fischer inner product, splitting over C and nonzero characteristic-zero scalar factors. ## External inputs and references ### [1] AldazRender — `external:AldazRender` Source lines 474–479. J.~M.~Aldaz and H.~Render, \href{https://doi.org/10.1007/s13324-023-00844-4}{A Fischer type decomposition theorem from the apolar inner product}, Analysis and Mathematical Physics 13 (2023), article~91. Fischer/apolar Hermitian inner-product construction employed in the local injectivity proof; its normalization and adjointness are supplied in the manuscript. Role: **used**. Expanded native manuscript lines 267–277. Source role only; the cited original work has not been independently checked for this neutral guide. Candidate references: `lem:shift` (external_reference). ### [2] Brion1993 — `external:Brion1993` Source lines 479–484. M.~Brion, \href{https://doi.org/10.1007/BF03026558}{Stable properties of plethysm: on two conjectures of Foulkes}, Manuscripta Mathematica 80 (1993), no.~4, 347--371. Earlier eventual surjectivity in the stabilization problem. Role: **background**. Expanded native manuscript lines 114–128. Source role only; the cited original work has not been independently checked for this neutral guide. - `sec:introduction` (background; source_audit_annotation): Earlier eventual surjectivity in the stabilization problem. Candidate references: `sec:introduction` (external_reference). ### [3] Brion — `external:Brion` Source lines 484–491. M.~Brion, \href{https://smf.emath.fr/publications/sur-certains-modules-gradues-associes-aux-produits-symetriques}{Sur certains modules gradu\'es associ\'es aux produits sym\'etriques}, in Alg\`ebre non commutative, groupes quantiques et invariants (Reims, 1995), S\'eminaires et Congr\`es 2, Soci\'et\'e Math\'ematique de France, Paris, 1997, 157--183. Earlier effective bound depending on degree and dimension. Role: **background**. Expanded native manuscript lines 114–128. Source role only; the cited original work has not been independently checked for this neutral guide. - `sec:introduction` (background; source_audit_annotation): Earlier effective bound depending on degree and dimension. Candidate references: `sec:introduction` (external_reference). ### [4] CIM — `external:CIM` Source lines 491–497. M.-W.~Cheung, C.~Ikenmeyer, and S.~Mkrtchyan, \href{https://doi.org/10.1016/j.jsc.2016.09.002}{Symmetrizing tableaux and the 5th case of the Foulkes conjecture}, Journal of Symbolic Computation 80 (2017), part~3, 833--843. \href{https://arxiv.org/abs/1509.03944v1}{arXiv:1509.03944v1}. Diagonal examples distinguishing specified canonical-map invertibility from ordinary representation comparison. Role: **background**. Expanded native manuscript lines 140–145. Source role only; the cited original work has not been independently checked for this neutral guide. - `sec:introduction` (background; source_audit_annotation): Diagonal examples distinguishing specified canonical-map invertibility from ordinary representation comparison. Candidate references: `sec:introduction` (external_reference). ### [5] Foulkes1950 — `external:Foulkes1950` Source lines 497–503. H.~O.~Foulkes, \href{https://doi.org/10.1112/jlms/s1-25.3.205}{Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form}, Journal of the London Mathematical Society 25 (1950), no.~3, 205--209. The ordinary Foulkes representation-comparison conjecture distinguished from canonical-map surjectivity. Role: **background**. Expanded native manuscript lines 130–139. Source role only; the cited original work has not been independently checked for this neutral guide. - `sec:introduction` (background; source_audit_annotation): The ordinary Foulkes representation-comparison conjecture distinguished from canonical-map surjectivity. Candidate references: `sec:introduction` (external_reference). ### [6] FultonHarris — `external:FultonHarris` Source lines 503–508. W.~Fulton and J.~Harris, \href{https://doi.org/10.1007/978-1-4612-0979-9}{Representation Theory: A First Course}, Graduate Texts in Mathematics 129, Springer-Verlag, New York, 1991. Classical weight-string interpretation of the injectivity criterion, which is proved locally. Role: **background**. Expanded native manuscript lines 252–258. Source role only; the cited original work has not been independently checked for this neutral guide. - `sec:introduction` (background; source_audit_annotation): Classical weight-string interpretation of the injectivity criterion, which is proved locally. Candidate references: `sec:differential` (external_reference). ### [7] Ikenmeyer — `external:Ikenmeyer` Source lines 508–513. C.~Ikenmeyer, On McKay's propagation theorem for the Foulkes conjecture, \href{https://arxiv.org/abs/1509.04957v1}{arXiv:1509.04957v1}, 2015. Earlier propagation/weight-shift argument compared with the manuscript’s two-sequence induction. Role: **background**. Expanded native manuscript lines 165–173. Source role only; the cited original work has not been independently checked for this neutral guide. - `sec:introduction` (background; source_audit_annotation): Earlier propagation/weight-shift argument compared with the manuscript’s two-sequence induction. Candidate references: `sec:introduction` (external_reference); `sec:differential` (external_reference). ### [8] Landsberg — `external:Landsberg` Source lines 513–520. J.~M.~Landsberg, \href{https://doi.org/10.1007/s11565-014-0202-7}{Geometric complexity theory: an introduction for geometers}, Annali dell'Universit\`a di Ferrara 61 (2015), no.~1, 65--117. \href{https://arxiv.org/abs/1305.7387}{arXiv:1305.7387}. Locators refer to the published version. Chow-variety interpretation and earlier polynomial-bound question. Role: **background**. Expanded native manuscript lines 107–128. Source role only; the cited original work has not been independently checked for this neutral guide. - `sec:introduction` (background; source_audit_annotation): Chow-variety interpretation and earlier polynomial-bound question. Candidate references: `sec:introduction` (external_reference). ### [9] Macdonald — `external:Macdonald` Source lines 520–525. I.~G.~Macdonald, \href{https://doi.org/10.1093/oso/9780198534891.001.0001}{Symmetric Functions and Hall Polynomials}, second edition, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1995. Complete reducibility used to choose an invariant complement in the representation corollary. Role: **used**. Expanded native manuscript lines 453–459. Source role only; the cited original work has not been independently checked for this neutral guide. - `cor:comparison` (proof_dependency; source_audit_annotation): The cited complete-reducibility result supplies the invariant complement used to split the surjection. Candidate references: `cor:comparison` (external_reference). ### [10] McKay — `external:McKay` Source lines 525–530. T.~McKay, \href{https://doi.org/10.1016/j.jalgebra.2007.12.003}{On plethysm conjectures of Stanley and Foulkes}, Journal of Algebra 319 (2008), no.~5, 2050--2071. Earlier propagation theorem described through Ikenmeyer’s formulation. Role: **background**. Expanded native manuscript lines 165–173. Source role only; the cited original work has not been independently checked for this neutral guide. - `sec:introduction` (background; source_audit_annotation): Earlier propagation theorem described through Ikenmeyer’s formulation. Candidate references: `sec:introduction` (external_reference). ### [11] RSW — `external:RSW` Source lines 530–536. C.~Raicu, S.~V.~Sam, and J.~Weyman, \href{https://doi.org/10.1007/s10013-021-00527-2}{On some modules supported in the Chow variety}, Vietnam Journal of Mathematics 50 (2022), no.~2, 501--521. \href{https://arxiv.org/abs/2108.10910v1}{arXiv:2108.10910v1}. Locators refer to this preprint version. Canonical-map/Chow-variety conventions and contextual normalization regularity; the local proof establishes its own diagonal criterion. Role: **background**. Expanded native manuscript lines 87–128. Source role only; the cited original work has not been independently checked for this neutral guide. - `sec:introduction` (background; source_audit_annotation): Canonical-map/Chow-variety conventions and contextual normalization regularity; the local proof establishes its own diagonal criterion. Candidate references: `sec:introduction` (external_reference); `sec:map` (external_reference). ## Coverage and limits {"mode": "native_latex_index", "claims": 5, "sections": 5, "external_references": 11, "source_bibliography_entries": 11, "unresolved_bibliography": 0, "unresolved": 0} - Reference edges are not certified proof dependencies. Edge direction is prerequisite → consumer. - Numbers reflect simple source counters; labels and line anchors are authoritative. - Extractive statements and proof openings are marked; they are not reconstructed proof gists. - Macros, included files and custom counters require an expanded source or annotated fallback.