Canonical bases via pairing monomials

TL;DR

This paper introduces a new formula and algorithm for computing the canonical basis in quantum groups of finite ADE type, with applications to affine Hecke algebra modules and simple module dimensions.

Contribution

It presents a novel formula for the bilinear form on monomials and an algorithm for canonical basis computation, extending to module dimensions.

Findings

New formula for the bilinear form on monomials in quantum groups

Algorithm for computing the canonical basis in finite ADE type

Extension of the algorithm to compute simple module dimensions

Abstract

For any quantum group of finite ADE type, we prove a new formula for the standard bilinear form evaluated at monomials. Combining this with ideas from the Lusztig-Shoji algorithm, we obtain a new algorithm that computes the canonical basis. In type A, the algorithm also computes composition multiplicities of standard modules for the affine Hecke algebra of $\text{GL}_n$ and we explain how the algorithm can be extended to compute the dimensions of simple modules.