[Paper Review] Mod-two cohomology of symmetric groups as a Hopf ring

Chad Giusti, Paolo Salvatore+1 more|arXiv (Cornell University)|Sep 17, 2009

Homotopy and Cohomology in Algebraic Topology20 references4 citations

TL;DR

This paper introduces a new additive basis and explicit multiplication rules for the mod-two cohomology of symmetric groups using a Hopf ring structure, with generators given by Thom classes of linear subvarieties. The key contribution is a complete, closed-form description of the cohomology ring via transfer products and Steenrod operations, unified through a geometric and invariant-theoretic framework that resolves longstanding complexity in cup product relations.

ABSTRACT

We compute the mod-2 cohomology of the collection of all symmetric groups as a Hopf ring, where the second product is the transfer product of Strickland and Turner. We first give examples of related Hopf rings from invariant theory and representation theory. In addition to a Hopf ring presentation, we give geometric cocycle representatives and explicitly determine the structure as an algebra over the Steenrod algebra. All calculations are explicit, with an additive basis which has a clean graphical representation. We also briefly develop related Hopf ring structures on rings of symmetric invariants and end with a generating set consisting of Stiefel-Whitney classes of regular representations v2. Added new results on varieties which represent the cocycles, a graphical representation of the additive basis, and on the Steenrod algebra action. v3. Included a full treatment of invariant theoretic Hopf rings, refined the definition of representing varieties, and corrected and clarified references.

Motivation & Objective

• To provide a complete, explicit presentation of the mod-two cohomology ring of symmetric groups, including multiplication and Steenrod operations.
• To establish a new additive basis using geometric and invariant-theoretic generators, represented by skyline diagrams.
• To unify and clarify previous approaches to symmetric group cohomology by leveraging a Hopf ring structure with transfer products.
• To give a closed-form description of the coproduct and Steenrod action using Dickson bi-partitions and Möbius inversion.
• To demonstrate that Stiefel-Whitney classes of standard representations generate the cohomology as a Hopf ring, with explicit coproduct formulas.

Proposed method

• The paper uses a Hopf ring structure on $ H^*(\coprod_n B\mathcal{S}_n; \mathbb{F}_2) $, with two multiplications: cup product and transfer product $\odot$, and a coproduct dual to the Pontrjagin product.
• Generators $\gamma_{\ell,n}$ are defined in degree $n(2^\ell - 1)$, with $\Delta\gamma_{\ell,n} = \sum_{i+j=n} \gamma_{\ell,i} \otimes \gamma_{\ell,j}$ and $\gamma_{\ell,n} \odot \gamma_{\ell,m} = \binom{n+m}{n} \gamma_{\ell,n+m}$.
• An additive basis is constructed using 'skyline diagrams', which generalize Young diagrams and encode monomials in the generators.
• The cup product is described via a combinatorial rule on skyline diagrams, analogous to symmetrized monomial multiplication.
• The Steenrod algebra action is computed via a Cartan formula for the transfer product, reducing the problem to computing $\text{Sq}^i(\gamma_{\ell,2^k})$.
• Stiefel-Whitney classes $w(k,\ell)$ are shown to generate the Hopf ring, with coproduct given by a sum over Dickson bi-partitions weighted by an $\mathbb{F}_2$-valued function $\phi$ defined via Möbius inversion.

Experimental results

Research questions

• RQ1 How can the mod-two cohomology of symmetric groups be described with a complete, explicit multiplication rule in a single basis?
• RQ2 What is the role of the Hopf ring structure—specifically the transfer product and coproduct—in organizing the cohomology ring?
• RQ3 Can geometric or invariant-theoretic generators (e.g., Thom classes, Stiefel-Whitney classes) be used to give a uniform description of the cohomology across all symmetric groups?
• RQ4 How does the Steenrod algebra act on the cohomology, and can this action be computed recursively from a small set of generators?
• RQ5 What is the precise coproduct formula for the Hopf ring generators, and how does it relate to combinatorial structures like Dickson bi-partitions?

Key findings

• The mod-two cohomology of $\coprod_n B\mathcal{S}_n$ is completely described as a Hopf ring generated by classes $\gamma_{\ell,n}$ with explicit relations: $\Delta\gamma_{\ell,n} = \sum_{i+j=n} \gamma_{\ell,i} \otimes \gamma_{\ell,j}$ and $\gamma_{\ell,n} \odot \gamma_{\ell,m} = \binom{n+m}{n} \gamma_{\ell,n+m}$.
• An additive basis is constructed using skyline diagrams, which provide a visual and combinatorial encoding of monomials in the generators, enabling explicit computation of cup products.
• The Steenrod algebra action is fully determined by the action on the generators $\gamma_{\ell,2^k}$, and a Cartan formula for the transfer product allows recursive computation of $\text{Sq}^i$ on arbitrary classes.
• The Stiefel-Whitney classes $w(k,\ell)$ of the standard representations generate the cohomology as a Hopf ring, with coproduct given by $\Delta w(k,\ell) = \sum_{p' \cup p'' \in \Pi_{k,\ell}} \phi(p' \cup p'') \left( \bigodot_{(k_i,\ell_i) \in p'} w(k_i,\ell_i) \right) \otimes \left( \bigodot_{(k_j,\ell_j) \in p''} w(k_j,\ell_j) \right)$.
• The function $\phi$ on Dickson bi-partitions is defined via Möbius inversion on the refinement poset, ensuring the coproduct formula pairs correctly with monomials in the $q$-generators.
• The paper shows that all cohomology classes are represented by Thom classes of linear subvarieties, and that restriction maps to elementary abelian subgroups are compatible with the Hopf ring structure.