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. 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. The Schur algebra S k ( n , r ) {displaystyle S_{k}(n,r)} can be defined for any commutative ring k {displaystyle k} and integers n , r ≥ 0 {displaystyle n,rgeq 0} . Consider the algebra k [ x i j ] {displaystyle k} of polynomials (with coefficients in k {displaystyle k} ) in n 2 {displaystyle n^{2}} commuting variables x i j {displaystyle x_{ij}} , 1 ≤ i, j ≤ n {displaystyle n} . Denote by A k ( n , r ) {displaystyle A_{k}(n,r)} the homogeneous polynomials of degree r {displaystyle r} . Elements of A k ( n , r ) {displaystyle A_{k}(n,r)} are k-linear combinations of monomials formed by multiplying together r {displaystyle r} of the generators x i j {displaystyle x_{ij}} (allowing repetition). Thus Now, k [ x i j ] {displaystyle k} has a natural coalgebra structure with comultiplication Δ {displaystyle Delta } and counit ε {displaystyle varepsilon } the algebra homomorphisms given on generators by Since comultiplication is an algebra homomorphism, k [ x i j ] {displaystyle k} is a bialgebra. One easilychecks that A k ( n , r ) {displaystyle A_{k}(n,r)} is a subcoalgebra of the bialgebra k [ x i j ] {displaystyle k} , for every r ≥ 0. Definition. The Schur algebra (in degree r {displaystyle r} ) is the algebra S k ( n , r ) = H o m k ( A k ( n , r ) , k ) {displaystyle S_{k}(n,r)=mathrm {Hom} _{k}(A_{k}(n,r),k)} . That is, S k ( n , r ) {displaystyle S_{k}(n,r)} is the linear dual of A k ( n , r ) {displaystyle A_{k}(n,r)} . It is a general fact that the linear dual of a coalgebra A {displaystyle A} 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 f {displaystyle f} , g {displaystyle g} on A {displaystyle A} , define their product to be the linear functional given by The identity element for this multiplication of functionals is the counit in A {displaystyle A} . Then the symmetric group S r {displaystyle {mathfrak {S}}_{r}} on r {displaystyle r} letters acts naturally on the tensor space by place permutation, and one has an isomorphism