Relaxation of strict parity for reducible Galois representations attached to the homology of \(\mathrm{GL}(3,\mathbb{Z})\) (Q2789370)

In this paper, the authors consider a reducible Galois representation of \(\mathrm{Gal}(\bar{\mathbb Q}/{\mathbb Q})\) over an algebraically closed field of characteristic \(p > 3\), which is a direct sum of a one-dimensional and a two-dimensional odd irreducible representations. Such a representation violates the strict parity condition of the generalization of Serre's conjecture made by the two authors and \textit{D. Pollack} in [Duke Math. J. 112, No. 3, 521--579 (2002; Zbl 1023.11025)]. The authors show that, under some conditions on the Serre conductor of the representation and the weight predicted by the conjecture, the representation still satisfies the conjecture, and thus the strict parity condition is not necessary for the conclusion of the conjecture.