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.
released 2026-09-25 | 1 theorem · 5 lemmas · 12 proofs · 9,818 words |
PLAY LEVEL 1 »(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 $\mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^b V)\hookrightarrow\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6 V)$.
released 2026-09-25 | 1 theorem · 1 lemma · 4 proofs · 2,914 words |
PLAY LEVEL 2 »(pdf)
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 $\dim V$.