# Two-row $W$-graphs in affine type $A$

## TL;DR

This paper constructs and proves the uniqueness of finite W-graphs for two-row shapes in affine symmetric groups, providing the first non-trivial examples in affine type and relating them to Lusztig's periodic W-graphs.

## Contribution

It introduces a new family of finite W-graphs in affine type A, expanding understanding of affine symmetric groups and their combinatorial structures.

## Key findings

- Constructed finite W-graphs for two-row shapes in affine type A
- Proved the uniqueness of these W-graphs
- Established an isomorphism with Lusztig's periodic W-graphs under positivity assumptions

## Abstract

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 examples 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.