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Foulkes' conjecture for sixth powers and quadratic stabilization

Proves the sixth case of Foulkes’ conjecture:

\mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^bV)

embeds equivariantly in

\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6V)

for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map

\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^aV)\to\mathop{\mathrm{Sym}}\nolimits ^a(\mathop{\mathrm{Sym}}\nolimits ^bV)

is surjective for a ≥ 2 and

b\ge a(a-1)

, giving dimension-independent quadratic stabilization.

Manuscripts in this family · 2

Foulkes' conjecture for the sixth symmetric power

Sep 25, 2026·main.pdf

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{\mathrm{GL}}\nolimits (V)

-equivariant injection

\mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^b V)\hookrightarrow\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6 V)

Full abstract

PDF on GitHub LaTeX source

Quadratic stabilization of the canonical Foulkes--Howe map

Sep 25, 2026·paper.pdfLean

For every finite-dimensional complex vector space V, we prove that the canonical Foulkes–Howe map

\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^a V)\longrightarrow\mathop{\mathrm{Sym}}\nolimits ^a(\mathop{\mathrm{Sym}}\nolimits ^b V)

is surjective whenever a ≥ 2 and

b\ge a(a-1)

. This gives a quadratic stabilization bound independent of

\dim V

Full abstract

PDF on GitHub LaTeX source