| Popis: |
Let G = GLn(k), the group of all invertible n x n matrices over an infinite field k. In this thesis we explore the cohomological relationship between a Schur algebra S(G) for G and the subalgebra S(B) corresponding to a Borel subgroup B of G. Our main motivation is the question of whether there is an analogue of the Kempf Vanishing Theorem in this setting. We place our study in a more general framework, defining subalgebras S(Ω,Г) of S(G) associated with certain intersections of parabolic subgroups of G, and investigate the connection between S(Ω,Г) and the subalgebra S(S(Ω,Ø). We define modules for S(Ω,Г) which serve as analogues for the Weyl modules for S(G). We produce bases for these Weyl modules and thereby show that S(Ω,Г) is a quasi- hereditary algebra. We find two-step projective presentations for the Weyl modules over subalgebras S(Ω,Г) of S(Ω,Г). and in special cases find projective resolutions. We use these to prove results which provide partial information on the existence of an analogue for the Kempf Vanishing Theorem, and on related questions. We derive a character formula for the Weyl modules which can be regarded as an extension of the Jacobi-Trudi identity for Schur functions. The methods used in this thesis are in the main elementary, with a heavy reliance on direct combinatorial arguments. |
