The author studied recently certain canonical bases for irreducible representations of quantum linear groups [5, 81. The basic ingredients in the present work are the Kazhdan-Lusztig bases and cells for Hecke and q-Schur algebras [14, 41. It has been proved that these canonical bases agree with Lusztig's canonical bases in the case of type A (see [8, 4.61). Compared to Lusztig's 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-James' 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, 71.
IC related bases for the coordinate algebra of a quantum matrix semi-group have been introduced in [5] (see also [7]), where the basis of "standard" monomials is the analogous T-basis while the IC basis B is the union of the dual Kazhdan-Lusztig bases for q-Schur algebras. This paper considers a similar problem corresponding to the coordinate algebra of a quantum linear group. We shall construct a basis fi for such a coordinate algebra via the IC basis B for the quantum matrix semi-group with the property that it gives rise to the canonical bases of irreducible corepresentations.
If this is the analogue of the 'C' basis for a Hecke algebra, what is the analogous basis corresponding to the 'T' basis? We shall show that an analogous basis may be defined via Dipper-Donkin's basis given in [2, (4.3.4)]. In fact, the basis fi and Dipper-Donkin's basis are quasi-IC related, that is, the transition matrices between them are the so-called quasi-pomatrices in the sense of [6, 1.21. Thus the theory in [6, 2.21 allows us to introduce a new basis T for the coordinate algebra of quantum linear groups such that fi and T are IC related. Taking the images of B and T, we shall obtain two bases Bo and To for the