In mathematical

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

, a good filtration is a

filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...

of a representation of a

reductive algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...

''G'' such that the

subquotient In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, thou ...

s are isomorphic to the spaces of

sections Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...

''F''(λ) of

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...

s λ over ''G''/''B'' for a

Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the 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. showed that the tensor product of two modules ''F''(λ)⊗''F''(μ) has a good filtration, completing the results of who proved it in most cases and who proved it in large characteristic. showed that the existence of good filtrations for these tensor products also follows from

standard monomial theory In algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive algebraic group by giving an explicit basis of elements called standard monomials. Many of th ...

References

* * * *{{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 ''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1 ...

, issn=0021-8693 , volume=77 , issue=1 , pages=162–185, doi-access=free Representation theory Linear algebraic groups