An algebraic model for rational U(2)-spectra

TL;DR

This work advances the explicit algebraic modeling of rational $G$-spectra for $G=U(2)$ by assembling seven blocks derived from toral and rank-2 toral data. By mapping through the central quotient $p:U(2)\to SO(3)$ and pulling back the seven-block partition from $SO(3)$, the authors obtain a decomposition of rational $U(2)$-spectra into five 1-dimensional and two 2-dimensional components, each dominated by the inverse image of a corresponding subgroup. Each block is described by sheaves of modules over subspace strata $\mathfrak{X}_{U(2)}$ with explicit ring and Weyl-group actions, inherited from the known $SO(3)$ models and adapted via fusion and normalizer data. The approach highlights how fusion, Weyl group dynamics, and central quotients shape the block structure and provide a concrete, computable framework for rational $U(2)$-spectra, extending the toral and rank-2 results and guiding future generalizations. Overall, the paper demonstrates a clear pathway to an algebraic model for rational $G$-spectra in a broad class of compact Lie groups.

Abstract

We construct an explicit and calculable models for rational U(2)-spectra. This is obtained by assembling seven blocks obtained in previous work: the toral part and earlier work on small toral groups. The assembly process requires detailed input on fusion and Weyl groups.

Theorems & Definitions (19)

• Example 3.1
• Example 3.2
• Lemma 4.1
• Lemma 4.2
• Lemma 5.1
• Corollary 5.2
• Lemma 5.3
• Lemma 5.4
• Lemma 5.5
• Corollary 5.6