Schur polynomial

From Wikipedia, the free encyclopedia

In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables with integral coefficients. The elementary symmetric polynomials and the complete homogeneous symmetric polynomials are special cases of Schur polynomials. Schur polynomials depend on a partition of natural number d into at most n parts, and are then homogeneous of degree d. The importance of Schur polynomials stems for a large part from their role in representation theory, notably of symmetric groups or of general linear groups, where they correspond to irreducible representations. The set of all Schur polynomials for fixed values of d form a linear basis for the space of all symmetric polynomials that are homogeneous of degree d. As a consequence every symmetric polynomial P can be written in a unique way as a linear combination of Schur polynomials; moreover if P has integral coefficients, then the linear combination has so as well. Any product of Schur functions can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood-Richardson rule.

Contents

[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

s_{(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, which can moreover be seen to be homogeneous of degree d.

[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 tableaux T of shape λ; the exponents t_1,\ldots,t_n give the weight of T, in other words each ti counts the occurrences of the number i in T.

[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

s_{(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)
s_{(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. s_{(2,1,1)} = e_1 \, e_3
  2. s_{(2,2,0)} = e_2^2 - e_1 \, e_3
  3. s_{(3,1,0)} = e_1^2 \, e_2 - e_2^2 - e_1 \, e_3
  4. s_{(4,0,0)} = e_1^4 - 3 \, e_1^2 \, e_2 + 2 \, e_1 \, e_3 + e_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

\phi = s_{(2,1,1)} - s_{(3,1,0)} + s_{(4,0,0)}.\,\!

[edit] Relation to representation theory

The Schur polynomials occur in the representation theory of the symmetric groups, 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