Monomials in canonical bases of quantum groups and quadratic forms

Abstract

We give an exact criterion for a monomial to belong to Lusztig's canonical basis for any quantised enveloping algebra corresponding to a symmetric Cartan datum. Using results of Marsh, we give a partial answer to a question of Lusztig (Israel Math. Conf. Proc. 7 (1993) 117–132). The key step is a combinatorial formula for the scalar product of two monomials in the quantised enveloping algebra.