Sums of 1-, 2- and 3-dimensional Galois representations and modular symbols for $\mathrm{GL}_n(\mathbb{Q})$

Let $\rho $ be an $n$-dimensional odd direct sum of irreducible mod $p$ odd Galois representations, each of dimension 1, 2, or 3. Assume that the 3-dimensional constituents satisfy the ADP conjecture, $p$ is sufficiently large, the Serre conductor $N$ of $\rho $ is squarefree, and the Serre conductors of the irreducible constituents of $\rho $ are all greater than 1. We prove that $\rho $ is attached to a Hecke eigenclass in the cohomology of $\varGamma _0(n,N)$ with coefficients in $M$, where $M$ is a finite-dimensional irreducible $\overline{\mathbb F}_p[\mathrm{GL}_n(\mathbb F_p)]$-representation $F$ tensored with a nebentype character $\epsilon $. The level $N$, the weight $F$, and the character $\epsilon $ are those predicted by a conjecture of Ash, Doud, and Pollack, which generalizes Serre’s conjecture for $\mathrm{GL}_2$. The proof uses modular symbols and their restriction to the Borel–Serre boundary.