Foulkes' conjecture for sixth powers and quadratic stabilization

1 · The essentials

These manuscripts compare two ways of building mathematical objects in two stages, reversing the sizes used at each stage. The question is whether everything built in one order can fit into what is built in the other, without losing information and while respecting the underlying symmetries. The first manuscript shows that such a fit exists when one size is six and the other is any whole number at least six. The second studies a particular rule going in the opposite direction and shows that it reaches every object on its destination side once the larger size is sufficiently large. Its threshold is the smaller size multiplied by one less than that size, and it does not depend on the size of the starting space. Both statements concern finite-dimensional spaces over the complex numbers. The second result has a Lean formalization; the supplied source does not identify the full sixth-power result as formalized.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine comparing six trays each holding space for b items with b trays each holding space for six items. The manuscripts concern whether two mathematical versions of these arrangements can be related without losing information, or whether a specified rule reaches every target arrangement—not merely whether the trays have the same total capacity.

Why it matters

The first manuscript settles, within its stated setting, the sixth-power case of the conjecture named in its title. The companion supplies a general threshold for the canonical map that works independently of the dimension of the starting space. Its Lean formalization provides formal verification for that stabilization result, rather than automatically verifying every statement in the family.

2 · Going further

For a finite-dimensional complex vector space $V$ , the first manuscript shows that, for every integer $b \geq 6$ , there is a $GL (V)$ -equivariant injection $Sym^{6} (Sym^{b} V) ↪ Sym^{b} (Sym^{6} V)$ . Here $Sym^{k}$ denotes the kth symmetric power. The injection preserves distinct inputs, while equivariance requires compatibility with the invertible linear transformations of $V$ . The source identifies this as the sixth-symmetric-power case of Foulkes’ conjecture.

The companion manuscript studies a specified map in the reverse direction: the canonical Foulkes–Howe map $Sym^{b} (Sym^{a} V) ⟶ Sym^{a} (Sym^{b} V)$ . Its Lean-formalized result states that this map is surjective for $a \geq 2$ and $b \geq a (a - 1)$ . Thus every element of the target has a preimage under this particular map. The threshold is quadratic in $a$ and independent of $dim V$ .

The two statements have related objects but different conclusions: one asserts the existence of an equivariant injection, and the other establishes surjectivity of a canonical map in the opposite direction. For $a = 6$ , the companion’s stated threshold is $b \geq 30$ , whereas the first manuscript’s injection statement starts at $b = 6$ . The abstracts supply these conclusions but no argument or construction, so they do not explain how the sixth-power result is obtained or how the companion is used in it.

Useful notions

Finite-dimensional complex vector spaces
Symmetric powers
Injective and surjective maps
Group actions and equivariance

What was known before / what is new

Before

The source frames the sixth-power statement as a case of Foulkes’ conjecture and identifies the canonical map as the Foulkes–Howe map. It gives no account of previously established cases, earlier bounds, or the history of the conjecture.

New

The first manuscript shows the equivariant embedding for every $b \geq 6$ and every finite-dimensional complex $V$ . The companion establishes surjectivity of the canonical map for all $a \geq 2$ and $b \geq a (a - 1)$ , giving a dimension-independent quadratic stabilization bound. The companion result has a Lean formalization.

Limits and assumptions

Both results assume a finite-dimensional complex vector space. The sixth-power embedding requires $b \geq 6$ ; the general canonical-map statement requires $a \geq 2$ and $b \geq a (a - 1)$ . The source does not state that this threshold is optimal or describe the map’s behavior below it. Formalization is explicitly attached to the quadratic stabilization manuscript, not to the full sixth-power embedding statement. No conclusion about all other cases of Foulkes’ conjecture, or a list of remaining open cases, is supplied.

Mini glossary

Symmetric power: The construction $Sym^{k} V$ combines k vector inputs with their order treated as irrelevant.
Equivariant injection: A map that preserves distinct inputs and respects the specified group actions.
Surjective map: A map for which every element of the target is the image of some input.
Foulkes–Howe map: The canonical map considered here from $Sym^{b} (Sym^{a} V)$ to $Sym^{a} (Sym^{b} V)$ .
Quadratic stabilization bound: Here, the threshold $b \geq a (a - 1)$ beyond which the canonical map is surjective, regardless of the dimension of V.

Keywords : Foulkes’ conjecture · Symmetric powers · Quadratic stabilization

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves the sixth case of Foulkes’ conjecture: $Sym^{6} (Sym^{b} V)$ embeds equivariantly in $Sym^{b} (Sym^{6} V)$ for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map $Sym^{b} (Sym^{a} V) \to Sym^{a} (Sym^{b} V)$ is surjective for a ≥ 2 and $b \geq a (a - 1)$ , giving dimension-independent quadratic stabilization.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/210.md

Manuscripts in this family

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Foulkes' conjecture for the sixth symmetric power

25/09/2026

To be verified

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 $GL (V)$ -equivariant injection $Sym^{6} (Sym^{b} V) ↪ Sym^{b} (Sym^{6} V)$ .

PDF GitHub folder

Quadratic stabilization of the canonical Foulkes--Howe map

25/09/2026

Verified by Lean

For every finite-dimensional complex vector space V, we prove that the canonical Foulkes–Howe map $Sym^{b} (Sym^{a} V) ⟶ Sym^{a} (Sym^{b} V)$ is surjective whenever a ≥ 2 and $b \geq a (a - 1)$ . This gives a quadratic stabilization bound independent of $dim V$ .

PDF GitHub folder Lean formalization