# Two families of commutation classes and categories attached to them

In this talk, I will introduce two families of commutation classes of the same

simply-laced finite type and their combinatorial features.

The one is called "adapted classes" and the another is called "twisted adapted classes".

Surprisingly, they encode information of categries of finite dimensional modules over Langlads dual quantum affine algebras in an interesting way.

Using the generalized Schur-Weyl daulity constructed by Kang-Kashiwara-Kim,

we can construct simplicity-preserving correspondences between "heart" subcategories of finite dimensional modules over Langlads dual quantum affine algebras in various ways. The correspondece between Langlads dual quantum affine algebras was obervated by Frenkel-Hernandez and is not well-understood yet. This is joint work with Kashiwara, Kim, Suh and

Scrimshaw.