Dual standard monomial theoretic and canonical bases for type \(A\) (Q1851329)

Let \(U_q^-(\mathfrak g)\) be the negative part of the quantized universal enveloping algebra constructed from a Cartan matrix associated to a complex semisimple Lie algebra \(\mathfrak g\) of type \(A\). Let \(\lambda\) be a dominant integral weight and \(V(\lambda)\) the irreducible \(U_q^-(\mathfrak g)\)-module with highest weight \(\lambda\). In this paper, the authors show that the duals of the standard monomial theoretic bases for various \(V(\lambda)\) patch together to give a so-called dual standard monomial theoretic basis for \(U_q^-(\mathfrak g)\). This basis lives in the crystal lattice and the image modulo \(q\) of a basis element is the corresponding standard tableau thought of as a crystal. The main result is that the transformation matrix between this basis and the canonical basis is unipotent upper triangular with respect to a natural order on the set of standard tableaux. And all this holds over the integral forms. Finally, they compute explicitly the dual standard monomial theoretic basis for \(U_q^-(sl_3)\).