Combine reductive quotient data with Brown’s algorithm for motivic periods

Develop a method that incorporates the reductive quotient of the motivic Galois group G_mot(M) into Brown’s algorithm (which computes linear relations among elements of the Hopf algebra of the unipotent radical via the coradical filtration) so as to obtain a complete description of linear relations among periods of a motive M in the hereditary Tannakian subcategory ⟨M⟩^.

Background

The paper connects dimension formulas for period spaces with structure theory of finite dimensional algebras and motives. It discusses Brown’s algorithmic approach for computing linear relations between motivic periods in hereditary Tannakian settings, which addresses the unipotent radical of the motivic Galois group.

However, a full analysis of all linear relations requires combining the unipotent radical with its reductive quotient. The authors note that existing methods do not yet integrate reductive quotient information into Brown’s framework, and examples indicate subtle behavior under this decomposition.

References

Under the assumption that $\langle M\rangle$ is hereditary, Brown explains an algorithm to describe the linear relations between the elements of the Hopf algebra of the unipotent radical of $G_{mot}(M)$ in terms of the coradical filtration. It remains open how to combine this with information on the reductive quotient.

— Dimension formulas for period spaces via motives and species (2405.21053 - Huber et al., 31 May 2024) in Remark rem:brown, Section 5 (Periods of motives)