Swapping symmetric powers: the sixth case and a quadratic bound

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Family 210 · Algebra
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-21

1. The question

Take symmetric products of vectors, then take symmetric products again. Does it matter which order the two degrees appear in?

For a complex vector space $V$, compare

$$\operatorname{Sym}^a(\operatorname{Sym}^b V) \quad\text{and}\quad \operatorname{Sym}^b(\operatorname{Sym}^a V).$$

They have the same total polynomial degree $ab$, but in higher dimensions they are generally different representations of $\operatorname{GL}(V)$. Foulkes’ conjecture predicts that when $a\leq b$, every irreducible type occurs at least as often in the second as in the first.

Family 210 contains two manuscripts dated 25 September 2026. M1 claims the full case $a=6$. M2 claims a uniform quadratic bound beyond which a particular canonical map in the reverse direction is surjective. The latter supplies the infinite range of M1’s main argument, leaving finitely many sixth-power comparisons.

2. Background and essential definitions

Symmetric powers and their symmetries

The symmetric power $\operatorname{Sym}^d V$ is formed from $d$-fold tensors by identifying tensors that differ only by permutation of their factors. If $V$ has a basis, its symmetric algebra can be viewed as a polynomial algebra in that basis. Products such as $v_1\cdots v_d$ are symmetric tensors.

Every invertible linear change of coordinates on $V$ acts on all these constructions. A map is $\operatorname{GL}(V)$-equivariant if applying the change of coordinates before or after the map gives the same result.

Over $\mathbb C$, finite-dimensional polynomial representations decompose into irreducibles called Schur modules $S_\lambda V$, indexed by partitions $\lambda$. A partition is a decreasing list of nonnegative integers; its size is their sum. A module with too many nonzero parts for $\dim V$ becomes zero.

Complete reducibility means that an equivariant injection exists precisely when every irreducible multiplicity in the source is at most its multiplicity in the target. Thus Foulkes’ conjecture is a whole family of inequalities, not simply a comparison of total dimensions.

Plethysm records composition

The character of a symmetric power is represented by a complete homogeneous symmetric function $h_d$. Composing symmetric-power operations corresponds to plethysm, written with brackets. The stable character of the first representation is $h_a[h_b]$.

The conjecture says that

$$h_b[h_a]-h_a[h_b]$$

is Schur-positive: its expansion in Schur functions $s_\lambda$ has nonnegative integer coefficients. These coefficients are the target multiplicities minus the source multiplicities.

One need not manipulate symmetric functions to understand the claim. They are a compact bookkeeping system for the simultaneous comparison of every irreducible type in every dimension.

A canonical map is a stronger requirement

Tensor positions can be arranged in a rectangle, regrouped after transposition, and symmetrized again. This produces a specified Foulkes–Howe map. M2 uses the orientation

$$\mu_{a,b,V}:\operatorname{Sym}^b(\operatorname{Sym}^a V) \longrightarrow\operatorname{Sym}^a(\operatorname{Sym}^b V).$$

If this map is surjective, complete reducibility gives an invariant complement to its kernel. Inverting the map on that complement supplies an equivariant injection in the direction predicted by Foulkes.

The injection obtained this way need not be a prescribed canonical map. That distinction matters even when $a=b$: the two representations then coincide, but a particular canonical endomorphism can still have a kernel.

3. Development history and previous partial results

Foulkes’ conjecture originated in his 1950 work on concomitants of forms. For two-dimensional $V$, classical Hermite reciprocity gives an isomorphism between the two compositions. The difficulty addressed here is uniformity in arbitrary dimension.

The case $a=1$ is immediate. Thrall’s decomposition formulas imply $a=2$, and Dent and Siemons proved $a=3$. Computations combined with McKay’s propagation theorem established the fourth and fifth cases: Müller and Neunhöffer supplied the relevant fourth-case computation, and Cheung, Ikenmeyer and Mkrtchyan supplied a fifth-case injectivity seed at $(5,6)$.

McKay’s theorem propagates canonical injectivity from a suitable seed to larger second parameters. The seed matters because canonical maps are not always invertible on the diagonal. Nonzero kernels were found at $(5,5)$ and $(6,6)$ in suitable dimensions, despite the ordinary representation comparison being automatic there.

Exact character calculations provided another route. Evseev, Paget and Wildon verified Foulkes’ multiplicity inequalities whenever the two parameters sum to at most $19$, covering the sixth case through $b=13$.

Brion proved eventual truth for fixed $a$, and later gave an effective dimension-dependent bound for surjectivity of the reverse canonical map. Geometrically, that map compares the coordinate ring of the Chow variety of decomposable forms with its normalization.

Raicu, Sam and Weyman developed the invariant-ring and normalization framework further. Their regularity bounds concern related graded modules, but do not by themselves identify the final nonzero degree of the quotient that controls canonical stabilization. M2 claims a sufficient bound depending only quadratically on $a$, independent of dimension. M1 combines its specialization at $a=6$ with finite multiplicity certificates.

4. What the new manuscripts claim

M1: the sixth symmetric power

For every integer $b\geq6$ and every finite-dimensional complex vector space $V$, M1 asserts an equivariant injection

$$\operatorname{Sym}^6(\operatorname{Sym}^b V) \hookrightarrow \operatorname{Sym}^b(\operatorname{Sym}^6 V).$$

Equivalently, $h_b[h_6]-h_6[h_b]$ is Schur-positive for every $b\geq6$. There is no restriction to a fixed dimension or stable-dimensional range. The diagonal case $b=6$ is immediate as a representation comparison.

M2: quadratic canonical stabilization

For every $a\geq2$, every $b\geq a(a-1)$, and every finite-dimensional complex $V$, M2 asserts that $\mu_{a,b,V}$ is surjective. The case $a=1$ is the identity map.

The bound is sufficient, not claimed to be sharp. It yields Foulkes’ multiplicity comparison throughout that range, but leaves the interval $a\leq b<a(a-1)$ untreated in general.

For $a=6$, the threshold is $30$. M1’s main route therefore needs finite comparisons only for $6\leq b\leq29$. It also records an independent route: its own chart and polarization argument gives canonical surjectivity for $b\geq150$, and its supplied programs cover the larger finite range through $149$. The latter route does not use M2’s quadratic theorem.

The finite computations are part of the manuscripts’ claimed arguments. They are reported here as source results, without a new execution or mathematical verification of the certificate programs.

5. Why the advance matters

Importance: 4/5, provisional. The family would settle a substantial new case of a longstanding plethysm conjecture and provide a dimension-independent polynomial stabilization bound for all outer degrees.

The two contributions complement each other. A sixth-case theorem addresses every second parameter for one fixed degree. The quadratic bound addresses every degree once the second parameter is sufficiently large. Neither alone is the full Foulkes conjecture, but together they reduce a broad asymptotic question to explicit finite ranges.

The canonical result also has geometric content. It bounds the degrees in which the coordinate ring of decomposable forms can differ from its normalization. This is more specific than knowing that the two nested symmetric powers have compatible dimensions or eventual multiplicities.

The rating reflects major progress with a clearly delimited scope. General multiplicity formulas, optimal stabilization thresholds and the remaining finite ranges for larger $a$ are separate questions.

6. The main idea

Turn surjectivity into a vanishing test

M2 realizes the target as equal-multidegree invariants in $(\operatorname{Sym}V)^{\otimes a}$. The canonical image is generated by degree-one invariants. More concretely, in degree $b$ it is spanned by tensors

$$P^{\otimes a},\qquad P=u_1u_2\cdots u_b,$$

where $P$ is a product of $b$ vectors.

A linear functional annihilating that image corresponds to a symmetric $a$-linear form $T$ on $\operatorname{Sym}^bV$ whose diagonal satisfies

$$T(P,\ldots,P)=0$$

for every such product. Surjectivity is equivalent to showing that this condition forces $T=0$.

Differentiate while retaining injectivity

For independent vector variables $U=(U_i)$ and $Y=(Y_i)$, consider the directional derivative

$$D_{Y,U}=\sum_iY_i\frac{\partial}{\partial U_i}.$$

It lowers the degree in $U$ by one and raises the degree in $Y$ by one. On polynomials homogeneous of degree $r$ in $U$ and degree $s$ in $Y$, the manuscript uses the standard positive-weight fact that this operator is injective when $r>s$.

The paper gives an inner-product version: with the Fischer inner product, multiplication and differentiation are adjoint, yielding

$$\|D_{Y,U}f\|^2-\|D_{U,Y}f\|^2=(r-s)\|f\|^2.$$

When $r>s$, a vanishing forward derivative therefore forces the original polynomial to vanish. Additional independent variables do not alter the criterion.

Factoring a binary form first produces a vanishing identity in which several slots share a common product. One sequence of directional derivatives frees the last slot. Induction on the number of slots then lets the other products vary independently. A second sequence removes their remaining common power.

The second sequence repeatedly raises the degree of the same destination variable. Keeping every derivative injective requires a degree budget, and that accumulation gives the sufficient inequality $b\geq a(a-1)$. The number of coordinates never enters the budget, explaining the dimension-independent bound.

Reduce the sixth case to six-part partitions

M1 denotes the source multiplicity at a partition $\lambda$ by $F(\lambda)$, and the target multiplicity by $Q(\lambda)$. The source has no constituents with more than six parts. It therefore suffices to compare those partitions; longer target constituents already have nonnegative multiplicity.

Stable character specialization allows the calculation in six variables to handle every dimension. Rectangular highest-weight shifts transfer known comparisons to new partitions. A chart calculation also certifies targets whose boxes outside the first row occupy at most $b$ positions.

The remaining finite argument combines upper bounds on $F$ with lower bounds on $Q$. Source bounds come from a signed-strip recurrence. Target bounds come from multiplying highest-weight polynomials with distinct leading monomials: distinct resulting monomials certify linear independence, so sumset estimates give many independent products.

After the broad certificates are applied, exact character differences are needed only in a narrow band of partition shapes. The manuscript computes that band using seven reused truncated rational-function numerators. It supplies exact programs, arithmetic bounds and reference outputs for the finite checks, rather than relying on floating-point approximations.

7. Scope, limitations, and open questions

The coefficients are complex. The proofs use complete reducibility, nonzero factorials in polarization, factorization of binary forms and a positive Hermitian inner product. No general positive-characteristic extension is asserted.

M1 proves existence of an equivariant injection. It does not claim injectivity of the canonical map in that direction for every sixth-case parameter. M2 proves surjectivity of a specified map in the opposite direction, only in its stated range. Keeping these orientations straight avoids turning a true multiplicity comparison into a stronger unsupported map assertion.

The quadratic bound is not presented as the least stabilization degree. For larger $a$, the finite interval below the bound still contains cases of the full conjecture. The manuscripts do not claim to resolve all of them.

The geometric normalization interpretation also concerns equality in sufficiently high graded pieces. It does not say that the coordinate ring and its normalization agree in every degree, or that the Chow variety is normal in all parameters.

8. Sources and further reading

Foulkes, Concomitants of the quintic and sextic up to degree four in the coefficients of the ground form, Journal of the London Mathematical Society 25 (1950), 205–209, is the original source. Brion, Stable properties of plethysm: on two conjectures of Foulkes, Manuscripta Mathematica 80 (1993), 347–371, supplies the eventual result.

Cheung, Ikenmeyer and Mkrtchyan, Symmetrizing tableaux and the 5th case of the Foulkes conjecture, Journal of Symbolic Computation 80 (2017), 833–843, gives the fifth-case computation and canonical-map distinctions. Ikenmeyer, On McKay’s propagation theorem for the Foulkes conjecture, explains propagation.

Raicu, Sam and Weyman, On some modules supported in the Chow variety, Vietnam Journal of Mathematics 50 (2022), 501–521, provides the invariant-ring framework. Landsberg, Geometric complexity theory: an introduction for geometers, discusses the polynomial-bound question answered by M2’s claim.

Source manuscripts

• M1. OpenAI, Foulkes’ conjecture for the sixth symmetric power (September 25, 2026).
• M2. OpenAI, Quadratic stabilization of the canonical Foulkes–Howe map (September 25, 2026).

Sources consulted on 21 October 2026 at repository revision 1581d4c. Download the manuscript citations.