Two-row W-graphs in affine type A

For affine symmetric groups we construct finite W-graphs corresponding to two-row shapes, and prove their uniqueness. This gives the first non-trivial family of purely combinatorial constructions of finite W-graphs in an affine type. We compare our construction with quotients of periodic W-graphs defined by Lusztig. Under certain positivity assumption on the latter the two are shown to be isomorphic.