Path combinatorics and light leaves for quiver Hecke algebras

We recast the classical notion of “standard tableaux" in an alcove-geometric setting and extend these classical ideas to all “reduced paths" in our geometry. This broader path-perspective is essential for implementing the higher categorical ideas of Elias–Williamson in the setting of quiver Hecke algebras. Our first main result is the construction of light leaves bases of quiver Hecke algebras. These bases are richer and encode more structural information than their classical counterparts, even in the case of the symmetric groups. Our second main result provides path-theoretic generators for the “Bott–Samelson truncation" of the quiver Hecke algebra.

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 [Formula: see text] 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 [Formula: see text], we obtain a positivity condition on some coefficients associated with the projection [Formula: see text] and the upper global basis, and prove several results related to the crystal bases. We then apply our results to finite type [Formula: see text] using the homogeneous simple modules [Formula: see text] indexed by one-column tableaux [Formula: see text].