Rational G-spectra for rank 2 toral groups of mixed type
J. P. C. Greenlees
TL;DR
This work constructs an explicit algebraic (abelian) model for the block of rational $G$-spectra over full subgroups when the identity component is a $2$-torus and the order‑2 component group acts nontrivially on $H_1(T)$, focusing on rank‑2 mixed components (three concrete group types). The approach parallels the torus case by realizing the sphere as a pullback of isotropically simple ring spectra, establishing a chain of Quillen equivalences to $DG$-$\mathcal A(G|\mathrm{full})$ via cellularization and isotropy separation, and proving a cellular skeleton theorem with injective dimension $2$. Central to the construction are an adelic cube of rings, inflation diagrams, Euler classes, and coinduced diagrams, which together translate the topological data into a precise abelian model and homology functors. The results provide a concrete, computable model for the toral blocks in the mixed two‑dimensional setting (notably resembling the normalizer of the maximal torus in $U(2)$) and set the stage for extending the algebraic modeling program to broader classes of rational $G$‑spectra.
Abstract
We give an explicit and calculable algebraic model for the block of rational G-spectra on full subgroups when G has identity component a 2-torus T, and component group of order 2 acting non-trivially on H_1(T). The example of particular interest is the normalizer of the maximal torus in U(2), which constitutes one of the most complicated blocks in the analysis of SU(3). This builds on the determination of subgroups up to conjugacy in arXiv 2501.06914