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. [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 and integers . Consider the algebra of polynomials (with coefficients in ) in commuting variables , 1 ≤ i, j ≤ . Denote by the homogeneous polynomials of degree . Elements of are k-linear combinations of monomials formed by multiplying together of the generators (allowing repetition). Thus
Now, has a natural coalgebra structure with comultiplication and counit the algebra homomorphisms given on generators by
Since comultiplication is an algebra homomorphism, is a bialgebra. One easily checks that is a subcoalgebra of the bialgebra , for every r ≥ 0.
Definition. The Schur algebra (in degree ) is the algebra . That is, is the linear dual of .
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 , on , 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 on 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.
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. [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 and integers . Consider the algebra of polynomials (with coefficients in ) in commuting variables , 1 ≤ i, j ≤ . Denote by the homogeneous polynomials of degree . Elements of are k-linear combinations of monomials formed by multiplying together of the generators (allowing repetition). Thus
Now, has a natural coalgebra structure with comultiplication and counit the algebra homomorphisms given on generators by
Since comultiplication is an algebra homomorphism, is a bialgebra. One easily checks that is a subcoalgebra of the bialgebra , for every r ≥ 0.
Definition. The Schur algebra (in degree ) is the algebra . That is, is the linear dual of .
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 , on , 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 on 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.