Computing Invariant Spaces via Global Cluster Analysis and Representation Theory

Abstract: The Singer algebraic transfer is a fundamental homomorphism in algebraic topology, providing a bridge between the homology of classifying spaces and the cohomology of the Steenrod algebra $\mathcal{A}$, which forms the $E_2$-term of the Adams spectral sequence. The domain of its dual is isomorphic to the space of $GL_k(\mathbb{F}_2)$-invariants in the quotient of the polynomial algebra, $(\mathcal{QP_k})^{GL_k(\mathbb{F}_2)}$, where $\mathcal{P}_k$ is regarded as a module over $\mathcal{A}$. A direct computation of this invariant space and its dual (i.e., the domain of the Singer transfer) remains a challenging problem. In this paper, we construct a new algorithm to compute $(\mathcal{QP_k})^{GL_k(\mathbb{F}_2)}$, which differs from the method presented in our recent work [15]. We refer to this new approach as the Global Cluster Analysis algorithm. It builds a \emph{weight interaction graph} to identify clusters of interacting weight spaces that form closed $\Sigma_k$-submodules (where $\Sigma_k \subset GL_k(\mathbb{F}_2)$). By performing invariance analysis on these larger clusters, our algorithm enables a complete and accurate computation of the global $\Sigma_k$-invariants, which are then used to determine the final $GL_k(\mathbb F_2)$-invariants. We also introduce an algorithm to directly compute the domain of the Singer transfer for ranks $k \leq 3$ in certain generic degrees, based entirely on Boardman's modular representation theory framework [2].