The Schur algebra S_k(n, r) can be defined for any
commutative ring k and
integers n, r \geq 0. Consider the
algebra k[x_{ij}] of
polynomials (with
coefficients in k) in n^2 commuting variables x_{ij}, 1 ≤
i,
j ≤ n. Denote by A_k(n, r) the homogeneous polynomials of
degree r. Elements of A_k(n, r) are
k-linear combinations of
monomials formed by multiplying together r of the generators x_{ij} (allowing repetition). Thus : k[x_{ij}] = \bigoplus_{r\ge 0} A_k(n, r). Now, k[x_{ij}] has a natural
coalgebra structure with comultiplication \Delta and counit \varepsilon the algebra homomorphisms given on generators by : \Delta(x_{ij}) = \textstyle\sum_l x_{il} \otimes x_{lj}, \quad \varepsilon(x_{ij}) = \delta_{ij}\quad (
Kronecker's delta). Since comultiplication is an algebra homomorphism, k[x_{ij}] is a
bialgebra. One easily checks that A_k(n, r) is a subcoalgebra of the bialgebra k[x_{ij}], for every
r ≥ 0.
Definition. The Schur algebra (in degree r) is the algebra S_k (n, r) = \mathrm{Hom}_k( A_k (n, r), k). That is, S_k(n,r) is the linear dual of A_k(n,r). It is a general fact that the linear
dual of a coalgebra 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 : \Delta(a) = \textstyle \sum a_i \otimes b_i and, given linear functionals f, g on A, define their product to be the linear functional given by : \textstyle a \mapsto \sum f(a_i) g(b_i). The identity element for this multiplication of functionals is the counit in A. == Main properties ==