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. (Mathieu 1990) showed that the tensor product of two modules F(λ)⊗F(μ) has a good filtration, completing the results of (Donkin 1985) who proved it in most cases and (Wang 1982) who proved it in large characteristic. (Littelmann 1992) showed that the existence of good filtrations for these tensor products also follows from standard monomial theory.
References
- Donkin, Stephen (1985), Rational representations of algebraic groups, Lecture Notes in Mathematics, 1140, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0074637, ISBN 978-3-540-15668-0
- Littelmann, Peter (1992), "Good filtrations and decomposition rules for representations with standard monomial theory", Journal für die reine und angewandte Mathematik 1992 (433): 161–180, doi:10.1515/crll.1992.433.161, ISSN 0075-4102
- Mathieu, Olivier (1990), "Filtrations of G-modules", Annales Scientifiques de l'École Normale Supérieure, Série 4 23 (4): 625–644, doi:10.24033/asens.1615, ISSN 0012-9593, http://www.numdam.org/item?id=ASENS_1990_4_23_4_625_0
- Wang, Jian Pan (1982), "Sheaf cohomology on G/B and tensor products of Weyl modules", Journal of Algebra 77 (1): 162–185, doi:10.1016/0021-8693(82)90284-8, ISSN 0021-8693
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