A stable splitting for spaces of commuting elements in unitary groups
Alejandro Adem, José Manuel Gómez, Simon Gritschacher
TL;DR
The paper proves a Miller-type stable splitting for spaces of commuting n-tuples in the unitary group U(m) after inverting m!, expressing Hom(Z^n,U(m))_+ as a wedge of Thom-like spaces tied to bundles of commuting varieties over generalized Grassmannians. This splitting is indexed by a poset 𝒫 of partitions and refined via a 𝒫-filtered stratification whose top terms involve the commuting varieties C_{|a|}(𝔲_{λ_a}). The authors extend the approach to symplectic and orthogonal groups and establish a parallel, though more delicate, splitting in those cases; they also prove a Steenrod-power obstruction showing the stable splitting does not hold integrally. An appendix studies the topology of the compactified commuting variety C_n(𝔤)^+, providing homology-sphere-type results that support the filtrations used. Overall, the work connects representations of abelian groups in classical groups with bundles over flag manifolds and commuting varieties, yielding a structured framework to analyze homological properties and indicating where integral stability fails due to Steenrod operations.
Abstract
We prove an analogue of Miller's stable splitting of the unitary group $U(m)$ for spaces of commuting elements in $U(m)$. After inverting $m!$, the space $\text{Hom}(\mathbb{Z}^n,U(m))$ splits stably as a wedge of Thom-like spaces of bundles of commuting varieties over certain partial flag manifolds. Using Steenrod operations we prove that our splitting does not hold integrally. Analogous decompositions for symplectic and orthogonal groups as well as homological results for the one-point compactification of the commuting variety in a Lie algebra are also provided.