Monoidal seeds of the categories ${\mathcal{C}}_{wv}$ over quiver Hecke algebras

Abstract

In this paper, when the quiver Hecke algebra R is symmetric, we present a new construction of quantum monoidal seeds ${\mathcal{S}}_{w,v}$ for ${\mathcal{C}}_{wv}$ using the reflection functors ${\mathcal{F}}_i$ and the newly introduced operators ${\mathcal{K}}_i$. The monoidal seed ${\mathcal{S}}_{w,v}$ is obtained as a subseed of the monoidal seed of ${\mathcal{C}}_w$ constructed by applying ${\mathcal{K}}_i$ and ${\mathcal{F}}_i$ along the special KF sequence determined by a reduced expression of $w$ and $v$. We further prove that the monoidal seed ${\mathcal{S}}_{w,v}$ coincides with the set of all prime factors of the determinantial modules $M(w_{\le k}{\Lambda }_{i_k},v_{\le k}{\Lambda }_{i_k})$. We prove that the Grothendieck ring $K({\mathcal{C}}_{wv})$ lies between the cluster algebra and the upper cluster algebra.

63 pages