Rational G-spectra for rank 2 toral groups of mixed type

TL;DR

This paper develops an explicit algebraic model for rational G-spectra when G has a rank 2 toral component with a non-trivial component group, aiding the analysis of complex group actions like in SU(3).

Contribution

It provides a new calculable algebraic framework for a specific class of rational G-spectra involving rank 2 toral groups with mixed type components.

Findings

Explicit algebraic model for rational G-spectra with rank 2 toral groups.

Application to the normalizer of the maximal torus in U(2).

Advances understanding of complex subgroup structures in 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