nLab
Schur function

Idea

The ring of symmetric polynomials in $n$ variables has a linear basis $\{s_{\lambda }\}$ of Schur polynomials indexed by partition?s $\lambda ={\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_n$ in $n$ parts.

The Schur polynomials are precisely the irreducible characters of finite dimensional polynomial representations of ${\mathrm{GL}}_n$. Also, the character ${\chi }_{\lambda }$ of $V_{\lambda }$, the irreducible representation of $S_k$ attached to $\lambda$ (for $k$ the size $\mid \lambda \mid$ of the partition) maps to the Schur polynomial under the character map $\mathrm{ch}$ from virtual characters to symmetric polynomials. This correspondance between representations of the symmetric groups and the general linear groups is called Schur Weyl Duality?

Definition

Given the partition ${\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_n$, the corresponding Schur polynomial is defined as follows. First define the $n\times n$-determinant (for any partition $\alpha$ in $n$ parts)

$$a_{\alpha }=\det (x_i^{{\alpha }_j}).$$a_\alpha = det (x_i^{\alpha_j}).

Let $\delta =(n-1,n-2,\dots ,1,0)$. Then the Schur polynomial attached to $\lambda$ is quotient

$$s_{\lambda }(x_1,x_2,\dots ,x_n)=a_{\lambda +\delta }/a_{\delta }.$$s_{\lambda}(x_1,x_2,\dots,x_n)=a_{\lambda+\delta}/a_{\delta}.

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 $s_{\lambda }$ for each partition, depending on countably many variables $x_1,x_2,\dots ,x_n,x_{n+1},\dots$.

Generalizations via Schur functors

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 $S_{\lambda }(V)\stackrel{S_{\lambda }(g)}{\to }{S}_{\lambda }(V)$ on an endomorphism $g\in \mathrm{GL}(V)$ is the Schur function of the eigenvalues of $g$. 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.

Literature

The authoritative monograph on the subject is

• I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Math. Monographs, 2nd enlarged ed. 1995

Other references

• Bruce E. Sagan, Schur functions, in (M. Hazewinkel, ed.) Encyclopaedia of Mathematics, Springer, pdf
• wikipedia
• Stuart Martin, Schur algebras and representation theory, Cambridge Univ. Press 1994

For generalizations see Jack polynomial and Macdonald polynomial.