Sums of two irreducible Galois representations and the homology of GLn(ℤ)

Given two irreducible Galois representations with relatively prime conductors, each attached to a Hecke eigenclass in an appropriate homology group, we prove that their direct sum is also attached to a Hecke eigenclass in a homology group, and that if the two Galois representations have weight, level, and nebentype predicted by a Serre-type conjecture of the authors and David Pollack, then so does the direct sum. Our methods utilize a study of [Formula: see text]-orbits of flags of subspaces of [Formula: see text], reducibility results for Galois representations attached to cohomology of parabolic subgroups of [Formula: see text], and a spectral sequence derived from the Tits building. In addition, we use the spectral sequence to prove two results about the degrees of homology to which irreducible Galois representations can be attached.