q-Schur algebras and quantized enveloping algebras
The main aim of this thesis is to investigate the relationship between the quantized enveloping algebra U(gln) (corresponding to the Lie algebra gln) and the q-Schur algebra, Sq(n, r). It was shown in [BLM] that there is a surjective algebra homomorphism θr : (gln)→Z[v, v -1] ⓍSq(n,r), where q = v2. §1 is devoted to background material. In §2, we show explicitly how to embed the q-Schur algebra into the r-th tensor power of a suitable n x n matrix ring. This gives a product rule for the q-Schur algebra with similar properties to Schur's product rule for the unquantized Schur algebra. A corollary of this is that we can describe, in §2.3, a certain family of subalgebras of the q-Schur algebra. In §3, we use the product rule of §2 to prove a q-analogue of Woodcock's straightening formula for codeterminants. This gives a basis of "standard quantized codeterminants" for Sq(n, r) which is heavily used in chapters 4, 5 and 6. In §4, we use the theory of quantized codeterminants developed in §3 to describe preimages under the homomorphism Or and the kernel of Or. In §5, we use the results of §3 and §4 to link the representation theories of U(gln} and Sq(n, r). We also obtain a simplified proof of Dipper and James' "semistandard basis theorem" for q- Weyl modules of q-Schur algebras. In §6, we show how to make the set of q-Schur algebras Sq(n, r) (for a fixed n) into an inverse system. We prove that the resulting inverse limit, Sv(n), is a cellular algebra which is closely related to the quantized enveloping algebra U(sln) and Lusztig's algebra U.