Friday, July 20, 14:00, 8.339
Dan Nakano (University of Georgia, Athens),
On Tensoring with the Steinberg Representation.
Let G be a simple, simply connected algebraic group over an
algebraically closed field of prime characteristic.
Recent work of Kildetoft and Nakano and of Sobaje has shown close connections
between two long-standing conjectures of Donkin: one on tilting modules
and the lifting of projective modules for Frobenius kernels of G and another
on the existence of certain filtrations of G-modules. A key question related
to these conjectures
is whether the tensor product of a Steinberg module with a simple module with
restricted highest weight admits a good filtration. In this talk, I will
survey results in this area and present new results
where we verify the aforementioned good filtration statement (i.e., Steinberg
tensored with restricted simple module) when
(i) p ≥ 2h-4
(h is the Coxeter number),
(ii) for all rank two groups,
(iii) for p ≥ 3 when the simple module corresponding to a
fundamental weight and
(iv) for a
number of cases when the rank is less than or equal to five.