nLab
Schur function

Idea

The ring of symmetric polynomials in nn variables has a linear basis {s λ}\{s_\lambda\} of Schur polynomials indexed by partitions λ=λ 1λ 2λ n\lambda=\lambda_1\geq \lambda_2\geq \ldots \geq \lambda_n in nn parts.

The Schur polynomials are precisely the irreducible characters of finite dimensional polynomial representations of GL nGL_n. Also, the character χ λ\chi_{\lambda} of V λV_\lambda, the irreducible representation of S kS_k attached to λ\lambda (for kk the size |λ||\lambda| of the partition) maps to the Schur polynomial under the character map chch 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 λ 1λ 2λ n\lambda_1\geq \lambda_2\geq \ldots \geq \lambda_n, the corresponding Schur polynomial is defined as follows. First define the n×nn\times n-determinant (for any partition α\alpha in nn parts)

a α=det(x i α j). a_\alpha = det (x_i^{\alpha_j}).

Let δ=(n1,n2,,1,0)\delta=(n-1, n-2, \dots, 1, 0). Then the Schur polynomial attached to λ\lambda is quotient

s λ(x 1,x 2,,x n)=a λ+δ/a δ. 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 λs_{\lambda} for each partition, depending on countably many variables x 1,x 2,,x n,x n+1,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 λ(V)S λ(g)S λ(V)\mathbf{S}_\lambda(V)\stackrel{\mathbf{S}_\lambda(g)}\to \mathbf{S}_\lambda(V) on an endomorphism gGL(V)g\in GL(V) is the Schur function of the eigenvalues of gg. 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, Macdonald polynomial, noncommutative Schur function, quasisymmetric Schur function.

Revised on March 5, 2014 02:34:49 by Zoran Škoda (161.53.130.104)