Global IC bases for quantum linear groups

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 Lusztig's canonical bases in the case of type A (see [8, 4.6]). 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, 7]. 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 B̃ 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 B̃ 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, 1.2]. Thus the theory in [6, 2.2] allows us to introduce a new basis T for the coordinate algebra of quantum linear groups such that B̃ and T are IC related. Taking the images of B̃ and T, we shall obtain two bases B̃ 0 and T 0 for the coordinate algebra of the quantum special linear group and prove that they are IC related too. A second observation of the paper is that our basis B̃ is nothing but the dual basis of B́, the canonical basis of the modified quantized enveloping algebra U introduced in [16, 17]. This is quite natural owing to the coincidence of the canonical bases for irreducible representations. Thus, we actually obtain a new (and dual) description for B in the case of type A.