A homotopy representation for a compact Lie group G is a type of G-CW complex. Fausk, Lewis, and May showed that there is an isomorphism between the Grothendieck group of homotopy representations and the group of invertible G-Spectra, Pic(G). This enabled them to describe Pic(G) using a left exact sequence. In order to refine the description of Pic(G), they conjectured that there is an isomorphism between the Grothendieck group of homotopy representations and that of generalized homotopy representations for G. If G is a finite group or a torus it is known that there is such an isomorphism. I will discuss my work toward proving that conjecture for any compact Lie group G.