Computing Invariant Spaces via Global Cluster Analysis and Representation Theory
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908480861896704 |
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| author | Phuc, Dang Vo |
| author_facet | Phuc, Dang Vo |
| contents | 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 $Σ_k$-submodules (where $Σ_k \subset GL_k(\mathbb{F}_2)$). By performing invariance analysis on these larger clusters, our algorithm enables a complete and accurate computation of the global $Σ_k$-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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04959 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computing Invariant Spaces via Global Cluster Analysis and Representation Theory Phuc, Dang Vo Algebraic Topology Rings and Algebras Representation Theory 55T15, 55S10, 55S05, 20C20, 68W30 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 $Σ_k$-submodules (where $Σ_k \subset GL_k(\mathbb{F}_2)$). By performing invariance analysis on these larger clusters, our algorithm enables a complete and accurate computation of the global $Σ_k$-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]. |
| title | Computing Invariant Spaces via Global Cluster Analysis and Representation Theory |
| topic | Algebraic Topology Rings and Algebras Representation Theory 55T15, 55S10, 55S05, 20C20, 68W30 |
| url | https://arxiv.org/abs/2508.04959 |