OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
Proves the sixth case of Foulkes’ conjecture: \mathop{Sym}\nolimits ^6(\mathop{Sym}\nolimits ^bV) embeds equivariantly in \mathop{Sym}\nolimits ^b(\mathop{Sym}\nolimits ^6V) for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map \mathop{Sym}\nolimits ^b(\mathop{Sym}\nolimits ^aV)→\mathop{Sym}\nolimits ^a(\mathop{Sym}\nolimits ^bV) is surjective for a ≥ 2 and b≥ a(a-1), giving dimension-independent quadratic stabilization.
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We prove the sixth-symmetric-power case of Foulkes' conjecture. For every integer b ≥ 6 and every finite-dimensional complex vector space V, there is a \mathop{GL}\nolimits (V)-equivariant injection \mathop{Sym}\nolimits ^6(\mathop{Sym}\nolimits ^b V)\hookrightarrow\mathop{Sym}\nolimits ^b(\mathop{Sym}\nolimits ^6 V).
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For every finite-dimensional complex vector space V, we prove that the canonical Foulkes–Howe map \mathop{Sym}\nolimits ^b(\mathop{Sym}\nolimits ^a V)\longrightarrow\mathop{Sym}\nolimits ^a(\mathop{Sym}\nolimits ^b V) is surjective whenever a ≥ 2 and b≥ a(a-1). This gives a quadratic stabilization bound independent of \dim V.
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