The ring of symmetric polynomials in variables has a linear basis of Schur polynomials indexed by partitions in parts.
The Schur polynomials are precisely the irreducible characters of finite dimensional polynomial representations of . Also, the character of , the irreducible representation of attached to (for the size of the partition) maps to the Schur polynomial under the character map from virtual characters to symmetric polynomials. This correspondance between representations of the symmetric groups and the general linear groups is called Schur Weyl Duality?
Given the partition , the corresponding Schur polynomial is defined as follows. First define the -determinant (for any partition in parts)
Let . Then the Schur polynomial attached to is quotient
As is usual in the theory of symmetric functions one can also deal with formal series in infinitely many variables. To make this precise one uses an inverse limit (see Macdonald) and obtains a Schur function for each partition, depending on countably many variables .
Schur functors may be viewed as a categorification of Schur functions. In fact, the Schur functors make sense in more general symmetric monoidal categories than vector spaces. It is a theorem in the case of vector space that the trace of a Schur functor on an endomorphism is the Schur function of the eigenvalues of . Considering the trace of a Schur functor makes sense in a general situation allowing for Schur functors and for the trace (rigid monoidal category); of course choosing appropriate variables to express the trace may depend on a context.
The authoritative monograph on the subject is
See also Schur positivity.