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  • Friday, July 20, 14:00, 8.339

    Dan Nakano (University of Georgia, Athens), On Tensoring with the Steinberg Representation.

  • Abstract:
    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.