A Specht Filtration of Permutation Modules Over KLR Algebras

In type A, Kleshchev-Ram-Mathas realize Specht modules as quotient of Permutation modules, in this paper, we construct a Specht filtration of Permutation modules indexed by hook partition in affine type A; and construct a generalized Specht filtration of Permutation modules indexed by any partition in linear quiver case.