Canonical bases via pairing monomials

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.