Algebra · Oct 6, 2026

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.

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