Double quiver algebras for non-cyclic chambers

Construct an extension of the double quiver algebras that represents non-cyclic chambers in which admissible crystal transitions may add or remove multiple atoms simultaneously, and determine the number and relations of the additional generators required.

Background

The double quiver algebra construction in the paper assumes cyclic chambers, where every allowed transition adds or removes one atom. In non-cyclic chambers, an intermediate one-atom configuration can be inadmissible even when a configuration obtained by adding several atoms is admissible. The authors propose generators directly connecting such configurations but do not construct or analyze the resulting algebra.

References

The details of this possible extension, such as the number of required new generators, etc., still require further explorations, and we leave the study on the non-cyclic chambers to future work.

— Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues  (2501.03365 - Bao et al., 6 Jan 2025) in Section 5.2, paragraph “General chambers”