Non-symmetric monoidal categorification of quantum unipotent coordinate rings

Establish that the initial monomial seed associated with a reduced expression of an element w in the Weyl group admits successive mutations in all directions, and thereby prove that the corresponding quiver Hecke module category provides a monoidal categorification of the quantum unipotent coordinate ring in the non-symmetric case.

Background

The paper uses quiver Hecke algebras to categorify quantum unipotent coordinate rings. For symmetric Cartan data, prior work shows that the initial monomial seed admits successive mutations in every direction, which yields a monoidal categorification. The analogous assertion for non-symmetric Cartan data is left unresolved and is relevant to extending the cluster-categorical framework beyond the symmetric setting.