Relative Duality and Structural Reductions for the Symmetric Hit Problem in Four Variables

Abstract

The symmetric hit problem asks whether the orbit sum of a monomial that is hit in $P(n)=\mathbf{F}_2[x_1,\ldots,x_n]$ is hit in the invariant subalgebra $B(n)=P(n)^{\Sigma_n}\cong H^*(BO(n);\mathbf{F}_2)$. The conjecture is known for $n\leq3$, while for $n\geq4$ nontrivial stabilizers impose equivariant constraints. Let $\bar{\iota}_d:Q_d(B(n))\to Q_d(P(n))$ be induced by inclusion, and define $\mathcal{C}_d(n)=\ker(\bar{\iota}_d)$. We prove a natural isomorphism \[ \mathcal{C}_d(n)^*\cong \frac{K_d(\mathrm{DS}(n))}{\rho_d(K_d(n))}, \] where $\mathrm{DS}(n)$ is the dual symmetric algebra and $\rho_d$ is restriction from the ordinary divided-power Steenrod kernel. Thus a symmetric polynomial already hit in $P(n)$ can represent a relative obstruction only through a symmetric Steenrod-kernel functional modulo those extending from $P(n)$. Combining this duality with a stabilizer-compatible coset-parity formula, we reduce the four-variable conjecture to a uniform local descent hypothesis for monomials with repeated exponents. We also prove an unconditional family. For every $t\geq0$, the ordered monomial $x^{\lambda_t}$ and the monomial symmetric function $m_{\lambda_t}$, where \[ \lambda_t=(10\cdot2^t-1,10\cdot2^t-1,2^{t+2}-1,2^{t+1}-1), \] are hit in $P(4)$ and $B(4)$, respectively, although $\omega(\lambda_t)>_l\omega_{\min}(|\lambda_t|)$. The base identity \[ m_{9,9,3,1}=\mathrm{Sq}^8(m_{5,5,3,1})+\mathrm{Sq}^1(m_{10,6,4,1}) \] follows from an exact Lucas--Cartan enumeration, and the resulting zero cohit classes propagate under the ordinary and symmetric Kameko isomorphisms. Exact linear algebra gives $\mathcal{C}_8(4)=\mathcal{C}_{12}(4)=\mathcal{C}_{14}(4)=\mathcal{C}_{22}(4)=0$. Finally, relative Kameko stability implies $\mathcal{C}_{26\cdot2^t-4}(4)=0$ for every $t\geq0$.