Abstract :
The author studied recently certain canonical bases for irreducible representations of quantum linear groups [5, 8]. The basic ingredients in the present work are the Kazhdan-Lusztig bases and cells for Hecke and q-Schur algebras [14, 4]. It has been proved that these canonical bases agree with Lusztigʹs canonical bases in the case of type A (see [8, 4.6]). Compared to Lusztigʹs definition of canonical bases, our construction in this case is more elementary and explicit in nature. This can also be seen from the link [8] of the canonical bases with Dipper-Jamesʹ semistandard bases. Their relation is just like the relation between the ‘T’ basis and the ‘C’ basis for a Hecke algebra. Such a relation or such a pair of bases has been referred to as an IC relation or IC related bases in [6, 7].
