A remark on convolution products for quiver Hecke algebras

TL;DR

This paper explores the relationship between convolution products in quiver Hecke algebras and tensor products in quantum groups, providing categorification, positivity results, and applications to type A finite cases.

Contribution

It offers a categorification of natural projections between dual highest weight modules via convolution products and establishes positivity and crystal basis results in symmetric cases.

Findings

Categorification of projection maps between dual highest weight modules.

Positivity conditions on coefficients related to global bases.

Application to finite type A using simple modules indexed by tableaux.

Abstract

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 $T$.