Kameko's Conjecture in Modular Invariant Theory

• Kameko's Conjecture is a central assertion in algebraic topology that provides explicit bounds on the dimensions of non-hit (cohit) polynomial spaces over F2.
• The conjecture employs a parameter-vector filtration and localized bounds linked to the action of the mod-2 Steenrod algebra on polynomial rings.
• Computational methods using systems like SageMath validate its predictions and reveal connections to algebraic transfers, homotopy types, and representation theory.

Kameko's conjecture is a central assertion in the algebraic topology of polynomial rings over the field $\mathbb F_2$, focusing on the action of the mod-2 Steenrod algebra on polynomials in several variables and the study of indecomposables in these modules. The conjecture provides explicit bounds on the dimensions of spaces of "non-hit" polynomials (“cohit” or indecomposable elements) in terms of group invariants, and has motivated new techniques in invariant theory, homological algebra, and computer algebra verification (Phuc, 31 Dec 2025).

1. Algebraic and Homological Framework

Let $\mathbb F_2$ denote the field with two elements. The mod-2 Steenrod algebra $\mathscr A$ is the graded $\mathbb F_2$-algebra generated by Steenrod squares $\mathrm{Sq}^k \in \mathscr A$, for $k \geq 0$, subject to the Adem relations and $\mathrm{Sq}^0=1$. For $P_m=\mathbb F_2[x_1,\ldots,x_m]$ with $|x_i|=1$, the algebra $P_m$ corresponds to $H^*\big((K(\mathbb F_2,1))^m;\mathbb F_2\big)$, naturally endowed with an unstable $\mathscr A$-module structure. The action is specified by

$\mathrm{Sq}^k(x_j^n) = \binom{n}{k} x_j^{n+k}$

and extended multiplicatively via the Cartan formula,

$$\mathrm{Sq}^k(fg) = \sum_i \mathrm{Sq}^i(f) \cdot \mathrm{Sq}^{k-i}(g).$$

The concept of "hit" (or $\mathscr A$-decomposable) polynomials, $f \in P_m$, refers to those expressible as $f \in \overline{\mathscr A} \cdot P_m$ for $$\overline{\mathscr A} = \oplus_{k>0} \mathrm{Sq}^k \mathbb F_2 \subset \mathscr A$$. The quotient $$Q^{\otimes m} := \mathbb F_2 \otimes_{\mathscr A} P_m \cong P_m / (\overline{\mathscr A} \cdot P_m)$$, and its degree-$n$ part $Q^{\otimes m}_n$, encapsulates the space of "cohit" or indecomposable polynomials. The Peterson hit problem seeks an explicit basis for these spaces.

2. Kameko’s Conjecture: Formulation and Parameter Filtration

2.1 Original (Global) Form

Kameko's conjecture, originally proposed in [M. Kameko, 1990], asserts a universal upper bound: $$\dim_{\mathbb F_2} Q^{\otimes m}_n \leq \prod_{j=1}^m (2^j - 1)$$ for all $m$ and $n$, bounding the total number of indecomposables in any degree by the product of $(2^j-1)$.

2.2 Localized Version (Parameter-Vector Bound)

Define the parameter vector $\mu=(\mu_1,\ldots,\mu_s)$ of a monomial $X=x_1^{u_1}\cdots x_m^{u_m}$ in terms of the binary expansions $u_j = \sum_k \epsilon_{k j} 2^k$, with $\mathrm{Param}_k = \sum_j \epsilon_{k-1,j}$. The localized variant conjectures: $$\dim_{\mathbb F_2} (Q^{\otimes m})^{\mu} \leq \prod_{j=1}^m (2^j-1)$$ where $(Q^{\otimes m})^{\mu}$ refers to the subquotient of $P_m$ of homogeneous degree $n$ with parameter vector $\mu$, modulo "hits" with lower $\mu$ in lexicographical order. The total degree $n$ is $\sum_k 2^{k-1} \mu_k$ (Phuc, 31 Dec 2025).

3. Main Results: The Fifth Algebraic Transfer and Validation

3.1 Generic Degrees and the Kameko Map

For $m=5$, define generic degrees as $n_s=5(2^t-1)+18 \cdot 2^t$ for $t \geq 0$. The Kameko map

$$\mathrm{Sq}^0_* : Q^{\otimes 5}_{n_s} \to Q^{\otimes 5}_{n_{s-1}}$$

is proven to be an isomorphism for $s>1$ (Wood–Kameko), reducing matters to explicit computation at $s=0,1$.

Explicitly,

• $\dim Q^{\otimes 5}_{n_0=18} = 730$;
• The kernel in $n_1$ has dimension $1900$;
• Consequently, for all $s \geq 1$, $\dim Q^{\otimes 5}_{n_s} = 730 + 1900 = 2630$.

3.2 Invariant Theory and Algebraic Transfer

The right action of $G(5)=\mathrm{GL}_5(\mathbb F_2)$ on $Q^{\otimes 5}_{n_s}$ yields a $2630$-dimensional module whose invariant space is $1$-dimensional. The explicit generator is constructed for $s=0,1$.

Singer’s algebraic transfer

$$\mathrm{Tr}_5: (\mathbb F_2 \otimes_{G(5)} P_{\mathscr A}P_5^*)_{n_s} \to \mathrm{Ext}_{\mathscr A}^{5,5+n_s}(\mathbb F_2, \mathbb F_2)$$

is shown to be an isomorphism in these bidegrees.

3.3 Validation in Low Degrees

Combined computations for $Q^{\otimes m}_d$ in all cases $d \leq 12$ and any $m \geq 1$ (summarized in Table 4.4 of (Phuc, 31 Dec 2025)) confirm the localized conjecture up to degree $12$, with explicit binomial-formulas (Corollary 4.4) utilized to check the upper bounds.

4. Technical Approach and Computational Verification

A systematic parameter‐vector filtration orders monomials by $\mu$ and refines the decomposition of $P_m$. The Wood–Kameko isomorphism (Theorem 2.3) provides isomorphisms for suitable $(m,n)$, reducing explicit computations to finite cases. Admissible (non-hit) monomials are detected using combinatorial methods, including spike and binary‐vector conditions, yielding finite candidate lists in each degree.

Explicit linear algebra is performed to write bases for $Q^{\otimes 5}_{n_0}$ and $Q^{\otimes 5}_{n_1}$ from admissible monomials, analyze the action of $\mathrm{Sq}^0_*$, and describe kernel and image structures. Invariant theory with respect to $G(5)$ relies on solving small linear systems, using permutation and transvection invariance to show 1-dimensionality of invariant subspaces.

All computations are fully implemented in the computer algebra systems SageMath and OSCAR, with verification, code, and output available in (Phuc, 14 Jul 2025) and (Phuc, 11 Sep 2025), as well as in the appendices and repositories linked from (Phuc, 31 Dec 2025).

5. Applications and Examples

5.1 Distinguishing Homotopy Types

By analyzing the action of the Steenrod algebra,

$\mathrm{Sq}^2 : H^6 \to H^8$

is nonzero for $H^6(\mathbb{C}P^4 / \mathbb{C}P^2)$ but zero for $H^6(\mathbb{S}^6\vee \mathbb{S}^8)$. Although their cohomology rings are isomorphic, their structures as $\mathscr A$-modules differ, and thus they are not homotopy equivalent (Phuc, 31 Dec 2025).

5.2 Representation Theory of $\mathrm{GL}_5(\mathbb F_2)$

In generic degrees $n_s$, $Q^{\otimes 5}_{n_s}$ stabilizes to a $2630$-dimensional module for the action of $G(5)$, with a 1-dimensional invariant ring. This provides an explicit link between algebraic topology, modular representation theory, and invariant theory.

5.3 Consequences for Cohit Decompositions and Higher Transfers

Explicit decompositions of $Q^{\otimes m}_d$ for $d \leq 12$ are tabulated (Table 4.4), confirming the localized Peterson bound. Extension of the method to the sixth algebraic transfer in bidegrees $(6,48\cdot 2^s)$ reveals that the map fails to hit a decomposable $\mathrm{Ext}$-element $h_5P\,h_2$ in bidegree $(6,48)$, but becomes an isomorphism in $(6,96)$.

6. Significance and Ongoing Directions

The results establish the validity of Kameko's conjecture and its localized variant in significant ranges, providing explicit structural data for indecomposables and resolving parts of the Peterson hit problem for $m=5$. The combination of parameter‐vector filtrations, advanced invariant theory, and explicit computer algebra verification represent a robust methodology, confirming theoretical predictions and yielding new algebraic insights (Phuc, 31 Dec 2025). The framework supports further study of higher transfers and cohit decompositions in modular invariant theory and unstable module categories.