Schur algebra

In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used to relate the representation theories of those two groups. Their use was promoted by the influential monograph of J. A. Green first published in 1980. The name "Schur algebra" is due to Green. In the modular case (over infinite fields of positive characteristic) Schur algebras were used by Gordon James and Karin Erdmann to show that the (still open) problems of computing decomposition numbers for general linear groups and symmetric groups are actually equivalent. Schur algebras were used by Friedlander and Suslin to prove finite generation of cohomology of finite group schemes.
Construction.
The Schur algebra formula_1 can be defined for any commutative ring formula_2 and integers formula_3. Consider the algebra formula_4 of polynomials (with coefficients in formula_2) in formula_6 commuting variables formula_7, 1 ≤ "i", "j" ≤ formula_8. Denote by formula_9 the homogeneous polynomials of degree formula_10. Elements of formula_9 are "k"-linear combinations of monomials formed by multiplying together formula_10 of the generators formula_7 (allowing repetition). Thus
Now, formula_4 has a natural coalgebra structure with comultiplication formula_16 and counit formula_17 the algebra homomorphisms given on generators by
Since comultiplication is an algebra homomorphism, formula_4 is a bialgebra. One easily
checks that formula_9 is a subcoalgebra of the bialgebra formula_4, for every "r" ≥ 0.
Definition. The Schur algebra (in degree formula_10) is the algebra formula_23. That is, formula_24 is the linear dual of formula_25.
It is a general fact that the linear dual of a coalgebra formula_26 is an algebra in a natural way, where the multiplication in the algebra is induced by dualizing the comultiplication in the coalgebra. To see this, let 
and, given linear functionals formula_28, formula_29 on formula_26, define their product to be the linear functional given by 
The identity element for this multiplication of functionals is the counit in formula_26.
Main properties.
Then the symmetric group formula_38 on formula_10 letters acts naturally on the tensor space by place permutation, and one has an isomorphism 
In other words, formula_24 may be viewed as the algebra of endomorphisms of tensor space commuting with the action of the symmetric group.
Generalizations.
The study of these various classes of generalizations forms an active area of contemporary research.