The transition matrix between PBW basis and semicanonical basis of U +( $\mathfrak{s}\mathfrak{l}_n $ (ℂ))

Abstract

We prove that the transition matrix between a special Poincare-Birkhoff-Witt (PBW) basis and the semicanonical basis of U+(\(\mathfrak{s}\mathfrak{l}_n \)(ℂ)) is upper triangular and unipotent under any order which is compatible with the partial order ⩽deg.