A remark on convolution products for quiver Hecke algebras

In this paper, we investigate a connection between convolution products for quiver Hecke algebras and tensor products for quantum groups. We give a categorification of the natural projection $\pi_{\lambda, \mu} : V_{\mathbb{A}}(\lambda)^\vee \otimes_{\mathbb{A}} V_\mathbb{A}(\mu)^\vee \twoheadrightarrow V_\mathbb{A}(\lambda+ \mu)^\vee$ sending the tensor product of the highest weight vectors to the highest weight vector in terms of convolution products. When the quiver Hecke algebra is symmetric and the base field is of characteristic $0$, we obtain a positivity condition on some coefficients associated with the projection $\pi_{\lambda, \mu}$ and the upper global basis, and prove several results related to the crystal bases. We then apply our results to finite type $A$ using the homogeneous simple modules $\mathcal{S}^T$ indexed by one-column tableaux $T$.