
School of Mathematics and Statistics
Alison Parker
University of Oxford
On the Weyl filtration dimension of the induced modules for a linear
algebraic group.
Friday 30th August, 121pm,
Carslaw 373.
Let G be a linear algebraic group over an algebraically closed field
of characteristic p whose corresponding root system is irreducible.
We denote the induced Gmodules, \nabla(\lambda) and the simple
Gmodules L(\lambda), for \lambda a dominant weight. A
Gmodule has a Weyl filtration if it is filtered by Weyl modules
(which are dual to the induced modules). Following Friedlander and
Parshall, we can measure how far a module is from having a Weyl
filtration by considering its Weyl filtration dimension (just as the
projective dimension measures in some sense how far a module is from
being projective). Since all projectives have a Weyl filtration these
two concepts are related. In fact we show how to calculate some
projective (and injective) dimensions for modules for the Schur
algebra using knowledge of the Weyl filtration dimension. The Weyl
filtration dimension has another advantage in that it is finite for a
finite dimensional Gmodule.
This talk will show how to calculate explicitly the Weyl filtration
dimension for the modules L(\lambda) and \nabla(\lambda) for
lambda a regular dominant weight. We then show how to deduce the
projective and injective dimensions for these modules considered as
modules for associated generalised Schur algebras (which includes the
usual Schur algebras for GL_{n}). We also deduce the global
dimension of the Schur algebras for GL_{n}, S(n,r), when
p>n and for S(mp,p) with m an integer.
