Schur polynomial

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In commutative algebra and invariant theory, Schur polynomials, named after Issai Schur, are certain homogeneous symmetric polynomials. They have the property that every symmetric polynomial in n variables that is homogeneous of degree d is a linear combination of certain Schur polynomials. More formally, there are finitely many Schur polynomials of degree d in n variables, and they form a linear basis of the homogeneous symmetric polynomials of degree d in n variables.

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[edit] Definition

Schur polynomials correspond to integer partitions. Given a partition

d = d_1 + d_2 + \cdots + d_n, \; \; d_1 \geq d_2 \geq \cdots \ge d_n

(where each dj is a non-negative integer), we can compute the corresponding Schur polynomial by expanding determinants

\sigma_{(d_1, d_2, \dots d_n)} (x_1, x_2, \dots x_n) = \frac{ \det \left[ \begin{matrix} x_1^{d_1+n-1} & x_2^{d_1+n-1} & \dots & x_n^{d_1+n-1} \\ x_1^{d_2+n-2} & x_2^{d_2+n-2} & \dots & x_n^{d_2+n-2} \\ \vdots & \vdots & \ddots & \vdots \\ x_1^{d_n} & x_2^{d_n} & \dots & x_n^{d_n} \end{matrix} \right]} { \det \left[ \begin{matrix} x_1^{n-1} & x_2^{n-1} & \dots & x_n^{n-1} \\ x_1^{n-2} & x_2^{n-2} & \dots & x_n^{n-2} \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \dots & 1 \end{matrix} \right].}

This gives a symmetric function because the numerator and denominator are each determinants which change sign under any transposition of the variables. Furthermore, the denominator is a Vandermonde determinant:

\Delta = \det \left[ \begin{matrix} x_1^{n-1} & x_2^{n-1} & \dots & x_n^{n-1} \\ x_1^{n-2} & x_2^{n-2} & \dots & x_n^{n-2} \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \dots & 1 \end{matrix} \right] = \prod_{1 \leq j < k \leq n} (x_j-x_k).

Each factor divides the determinant in the numerator, so the quotient is a polynomial.

[edit] Properties

Because we can readily enumerate the distinct partitions of d into n parts using Ferrers diagrams, using this formula we can write down all the degree d Schur polynomials in n variables, giving a linear basis for the space of homogeneous degree d symmetric polynomials in n variables.

Each Schur polynomial in n variables is a polynomial function of the elementary symmetric polynomials

e_0 (x_1, x_2, \dots, x_n) = 1,
e_1 (x_1, x_2, \dots, x_n) = \sum_{1 \leq j \leq n} x_j,
e_2 (x_1, x_2, \dots, x_n) = \sum_{1 \leq j < k \leq n} x_j \, x_k,

and so forth, down to

e_n (x_1, x_2, \dots, x_n) = x_1 \, x_2 \cdots x_n.

Explicit expressions can be found using computational techniques from elimination theory, perhaps the most elementary of which are Gröbner bases using an elimination order.

For a partition λ, the Schur function can be expanded as a sum of monomials as

s_\lambda(x_1,x_2,\ldots,x_n)=\sum_T x_1^{t_1}\cdots x_n^{t_n}

where the summation is over all semistandard Young tableau of shape λ and content 1,2,\ldots,n in which the number i appears ti times.

[edit] Example

The following extended example should help clarify these ideas. Consider the case n = 3, d = 4. Using Ferrers diagrams or some other method, we find that there are just four partitions of 4 into at most three parts. We have

\sigma_{(2,1,1)} (x_1, x_2, x_3) = \frac{1}{\Delta} \; \det \left[ \begin{matrix} x_1^4 & x_2^4 & x_3^4 \\ x_1^2 & x_2^2 & x_3^2 \\ x_1 & x_2 & x_3 \end{matrix} \right] = x_1 \, x_2 \, x_3 \, (x_1 + x_2 + x_3)
\sigma_{(2,2,0)} (x_1, x_2, x_3) = \frac{1}{\Delta} \; \det \left[ \begin{matrix} x_1^4 & x_2^4 & x_3^4 \\ x_1^3 & x_2^3 & x_3^3 \\ 1 & 1 & 1 \end{matrix} \right]= x_1^2 \, x_2^2 + x_1^2 \, x_3^2 + x_2^2 \, x_3^2 + x_1^2 \, x_2 \, x_3 + x_1 \, x_2^2 \, x_3 + x_1 \, x_2 \, x_3^3

and so forth. Summarizing:

  1. \sigma_{(2,1,1)} = \alpha_1 \, \alpha_3
  2. \sigma_{(2,2,0)} = \alpha_2^2 - \alpha_1 \, \alpha_3
  3. \sigma_{(3,1,0)} = \alpha_1^2 \, \alpha_2 - \alpha_2^2 - \alpha_1 \, \alpha_3
  4. \sigma_{(4,0,0)} = \alpha_1^4 - 3 \, \alpha_1^2 \, \alpha_2 + 2 \, \alpha_1 \, \alpha_3 + \alpha_2^2

Every homogeneous degree-four symmetric polynomial in three variables can be expressed as a unique linear combination of these four Schur polynomials, and this combination can again be found using a Gröbner basis for an appropriate elimination order. For example,

\phi(x_1, x_2, x_3) = x_1^4 + x_2^4 + x_3^4

is obviously a symmetric polynomial which is homogeneous of degree four, and we have

φ = σ(2,1,1) − σ(3,1,0) + σ(4,0,0).

[edit] Relation to representation theory

The Schur polynomials occur in the representation theory of the general linear groups and unitary groups, and in fact this is how they arose. The Weyl character formula helps to generalize Schur's work to other compact and semisimple Lie groups.

[edit] See also

[edit] References

  • Sturmfels, Bernd (1993). Algorithms in Invariant Theory. New York: Springer. ISBN 0-387-82445-6.  is a beautiful introduction to computational methods in invariant theory.
  • Tignol, Jean-Pierre (2001). Galois's Theory of Algebraic Equations. Singapore: World Scientific. ISBN 981-02-4541-6.  Offers some nice historical background.
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