good filtration
In mathematical representation theory, a good filtration is a filtration of a representation of a reductive algebraic group G such that the subquotients are isomorphic to the spaces of sections F(λ) of line bundles λ over G/B for a Borel subgroup B. In characteristic 0 this is automatically true as the irreducible modules are all of the form F(λ), but this is not usually true in positive characteristic. {{harvtxt|Mathieu|1990}} showed that the tensor product of two modules F(λ)⊗F(μ) has a good filtration, completing the results of {{harvtxt|Donkin|1985}} who proved it in most cases and {{harvtxt|Wang|1982}} who proved it in large characteristic. {{harvtxt|Littelmann|1992}} showed that the existence of good filtrations for these tensor products also follows from standard monomial theory.
References
- {{Citation | last1=Donkin | first1=Stephen | title=Rational representations of algebraic groups | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-15668-0 | doi=10.1007/BFb0074637 |mr=804233 | year=1985 | volume=1140}}
- {{Citation | last1=Littelmann | first1=Peter | authorlink = Peter Littelmann | title=Good filtrations and decomposition rules for representations with standard monomial theory | doi=10.1515/crll.1992.433.161 |mr=1191604 | year=1992 | journal=Journal für die reine und angewandte Mathematik | issn=0075-4102 | volume=1992 | issue=433 | pages=161–180| s2cid=116470877 }}
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- {{Citation | last1=Wang | first1=Jian Pan | title=Sheaf cohomology on G/B and tensor products of Weyl modules | doi=10.1016/0021-8693(82)90284-8 |mr=665171 | year=1982 | journal=Journal of Algebra | issn=0021-8693 | volume=77 | issue=1 | pages=162–185| doi-access=free }}