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.
Green first published in 1980.
[1] 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.
[2] Schur algebras were used by Friedlander and Suslin to prove finite generation of cohomology of finite group schemes.
[3] The Schur algebra
can be defined for any commutative ring
of polynomials (with coefficients in
commuting variables
the homogeneous polynomials of degree
are k-linear combinations of monomials formed by multiplying together
(allowing repetition).
has a natural coalgebra structure with comultiplication
the algebra homomorphisms given on generators by Since comultiplication is an algebra homomorphism,
One easily checks that
is a subcoalgebra of the bialgebra
The Schur algebra (in degree
It is a general fact that the linear dual of a coalgebra
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
, define their product to be the linear functional given by The identity element for this multiplication of functionals is the counit in
Then the symmetric group
letters acts naturally on the tensor space by place permutation, and one has an isomorphism In other words,
may be viewed as the algebra of endomorphisms of tensor space commuting with the action of the symmetric group.
The study of these various classes of generalizations forms an active area of contemporary research.