Math RefereeQuadratic stabilization of the canonical Foulkes–Howe map
Differential injectivity and a quadratic surjectivity range
The proposed argument
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.
The reading guide describes the manuscript’s argument. Independent human assessment remains pending. Review assessments and source checks are available separately. The map indexes four proof-bearing results and a numbered remark; the unnumbered annihilator criterion sits in its section.
Reading order
- 01
Specify the canonical image
Use averaging and multiplication to define the map and its image, then express surjectivity through an annihilator.
- 02
Establish one-step injectivity
Use adjoint directional derivatives and a norm identity on homogeneous spaces.
- 03
Run the slot induction
Combine binary splitting, two derivative sequences and the induction hypothesis to obtain multilinear vanishing.
- 04
Derive the representation comparison
Split the surjection equivariantly and state the scope of the construction.
Follow the dependencies
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Your independent assessment
Identify the direction and normalization of the canonical map. Check the commutator sign, both differential degree ledgers and the endpoint b = a(a−1). Distinguish the main surjectivity theorem from the splitting used in the corollary.