Existence of integral equivariant Moore spectra for arbitrary irreducible representations

Determine whether, for every irreducible complex representation ρ of a finite group G, there exists an integral equivariant Moore spectrum M(ρ), i.e., a G-CW spectrum whose integral homology is concentrated in degree 0 and whose H_0 carries the given representation ρ as the induced G-action.

Background

Equivariant Moore spectra play a central role in the paper’s approach, providing coefficient objects in equivariant algebraic K-theory that mirror Galois representations. The authors construct such spectra in many cases (e.g., sums of induced abelian characters), but note a gap for general irreducible representations.

Resolving existence in full generality would remove a key technical assumption and enable the proposed equivariant Quillen–Lichtenbaum framework to apply to arbitrary higher-dimensional Galois representations.