Computational Approaches to the Singer Transfer: Preimages in the Lambda Algebra and $G_k$-Invariant Theory

TL;DR

This paper introduces an algorithmic method using the lambda algebra and invariant theory to compute preimages under the Singer transfer, revealing errors in previous proofs and providing explicit preimages and a SageMath tool for automating these computations.

Contribution

It develops a systematic, linear algebra-based approach to compute preimages in the Singer transfer, corrects prior results, and provides a complete SageMath implementation for invariant space calculations.

Findings

Disproved the previous proof regarding the element d_0 in Ext groups.

Explicitly determined a preimage for p_0 in Ext groups.

Developed a SageMath algorithm for automating invariant space computations.

Abstract

We present a systematic, algorithmic method to compute the preimage of elements under the Singer algebraic transfer. Using the lambda algebra and the invariant-theoretic formula of P.H. Chon and L.M. Ha [5], we formulate the preimage search as a solvable problem in linear algebra. This framework is applied to study key indecomposable elements in the Adams spectral sequence. As a consequence, we show that the proof of the known result that the indecomposable element $d_0 \in \mathrm{Ext}^{4,18}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2).$ lies in the image of the fourth Singer transfer, as given by Nguyen Sum in [17], is false. Furthermore, we provide the explicit description of a preimage for the indecomposable element $p_0 \in \mathrm{Ext}^{4,37}_{\mathcal A}(\mathbb Z/2, \mathbb Z/2).$ This preimage had not been explicitly determined in the previous work of N.H.V. Hung and V.T.N. Quynh [8]. Finally, our most significant contribution is the construction of a complete \textsc{SageMath} algorithm that fully automates the computation of both the dimension and an explicit basis for the $G_k$-invariant space $[(Q\mathcal{P}_k)_d]^{G_k}$. This tool facilitates the verification of our results [11, 12, 13] that were previously computed manually in connection with Singer's conjecture for rank 4.