Let $\mathfrak{g}_{0}$ be a simple Lie algebra of type ADE and let $U'_{q}(\mathfrak{g})$ be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group $B(\mathfrak{g}_{0})$ on the quantum Grothendieck ring $\mathcal{K}_{t}(\mathfrak{g})$ of Hernandez-Leclerc’s category $\mathcal{C}_{\mathfrak{g}}^{0}$. Focused on the case of type $A_{N-1}$, we construct a family of monoidal autofunctors $\{\mathcal{S}_{i}\}_{i\in \mathbf{Z}}$ on a localization $\mathcal{T}_{N}$ of the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty}$. Under an isomorphism between the Grothendieck ring $K(\mathcal{T}_{N})$ of $\mathcal{T}_{N}$ and the quantum Grothendieck ring $\mathcal{K}_{t}(A^{(1)}_{N-1})$, the functors $\{\mathcal{S}_{i}\}_{1\leq i\leq N-1}$ recover the action of the braid group $B(A_{N-1})$. We investigate further properties of these functors.

Sufficiency of jets is a very important notion introduced by René Thom in order to establish the structural stability theory. The criteria for some sufficiency of jets are known as the Kuo condition and Thom type inequality, which are defined using the Kuo quantity and Thom quantity. Therefore these quantities are meaningful. In this paper we show the equivalence of Kuo and Thom quantities. Then we apply this result to the relative conditions to a given closed set.