Orbit-Type Structure and a Counterexample to Singer's Conjecture for the Sixth Algebraic Transfer

Abstract

Let $\mathscr A$ be the Steenrod algebra over the field of characteristic two, $\mathbb F_2$, and let $GL(q)$ be the general linear group over $\mathbb F_2$. The algebraic transfer introduced by Singer relates modular invariant theory for $P_q=\mathbb F_2[x_1,\ldots,x_q]$ to the cohomology groups $\operatorname{Ext}_{\mathscr A}^{q,*}(\mathbb F_2,\mathbb F_2)$. William Singer conjectured that this transfer is always a monomorphism. This conjecture has stood for nearly 40 years, and in this work we demonstrate that it fails in general. Specifically, we disprove the conjecture in bidegree $(6,6+36)$ by computing $[(QP_6)_{36}]^{GL(6)}$, where $QP_6=\mathbb F_2\otimes_{\mathscr A}P_6$. Moving beyond standard algorithmic verification, we introduce a deterministic post-computational analytical procedure to investigate the $Σ_6$-orbit structure of the reduced representatives. While the degree-$15$ target invariant naturally decomposes into quasisymmetric blocks, the two degree-$36$ kernel invariants are strictly classified by their support slices, full orbit sizes, and stabilizer isomorphism types. By decoupling the exact algebraic solution from its symmetric-group footprint, this approach transforms opaque raw computational data into structurally explicit combinatorial invariants.