Graphs arising from the dual Steenrod algebra
Connor Elliott, Courtney Hauf, Kai Morton, Sarah Petersen, Leticia Schow
TL;DR
This work extends Wood's graph-theoretic interpretation of the mod $2$ dual Steenrod algebra to odd primes and to the $C_2$-equivariant setting, encoding monomials in quotients $A^ ext{vee}(n)$ and $A_{C_2}^ ext{vee}(n)$ as graphs with carefully chosen vertex sets. It provides explicit connectedness criteria using combinatorial sums $C(r,s)$ (and a unilateral variant with $U(r,s)$) derived from the $p$-adic (or 2-adic) expansions of monomial exponents, and it characterizes trees and Hamilton cycles via digit-sum conditions and degree bounds; it also gives graph-theoretic interpretations of coproduct and antipode for generators in these Hopf (algebroid) structures. In addition to the odd-prime and equivariant extensions, the paper discusses open questions, including potential extensions to comodules and deeper algebraic interpretations of the Hopf structures through graphs, aiming to aid Adams spectral sequence computations. Overall, the results fuse combinatorial graph theory with Hopf-algebraic data from unstable and equivariant Steenrod algebras, offering new intuition and tools for homotopy-theoretic computations and potential future generalizations.
Abstract
We extend Wood's graph theoretic interpretation of certain quotients of the mod $2$ dual Steenrod algebra to quotients of the mod $p$ dual Steenrod algebra where $p$ is an odd prime and to quotients of the $C_2$-equivariant dual Steenrod algebra. We establish connectedness criteria for graphs associated to monomials in these algebra quotients and investigate questions about trees and Hamilton cycles in these settings. We also give graph theoretic interpretations of algebraic structures such as the coproduct and antipode arising from the Hopf algebra structure on the mod $p$ dual Steenrod algebra and the Hopf algebroid structure of the $C_2$-equivariant dual Steenrod algebra.