Path combinatorics and light leaves for quiver Hecke algebras

TL;DR

This paper introduces a new path-based approach to quiver Hecke algebras, constructing light leaves bases and path-theoretic generators that enhance structural understanding and extend classical combinatorial concepts.

Contribution

It extends classical tableaux to a geometric path setting and constructs richer light leaves bases for quiver Hecke algebras, advancing higher categorical methods.

Findings

Constructed light leaves bases for quiver Hecke algebras.

Provided path-theoretic generators for Bott--Samelson truncations.

Enhanced structural understanding of quiver Hecke algebras.

Abstract

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.