Kleshchev–Mathas–Ram give a presentation of the Specht module \(S^\lambda \) as a quotient of the permutation module \(M^\lambda \) . In this paper, we construct a (graded) Specht filtration of the permutation module \(M^\lambda \) in the following cases: when \(\lambda = (k,1^r)\) is a hook partition, over the KLR algebra of type \(A^{(1)}_{e-1}\) for \(e > 2\) ; and when \(\lambda = (k,r)\) is a two-row partition with \(k \ge r\) , over the KLR algebra of type \(A_\infty \) . Furthermore, when \(\lambda \) is an arbitrary partition in type \(A_\infty \) , we construct a filtration of \(M^\lambda \) such that each subquotient \(M_i / M_{i+1}\) admits a Specht resolution.
