Schur Polynomial - Definition

Definition

Schur polynomials correspond to integer partitions. Given a partition

(where each d_j is a non-negative integer), the following functions are alternating polynomials (in other words they change sign under any transposition of the variables):

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

Since they are alternating, they are all divisible by the Vandermonde determinant:

$a_{(n-1, n-2, \dots, 0)} (x_1, x_2, \dots, x_n) = \det \left[ \begin{matrix} x_1^{n-1} & x_2^{n-1} & \dots & x_n^{n-1} \\ x_1^{n-2} & x_2^{n-2} & \dots & x_n^{n-2} \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \dots & 1 \end{matrix} \right] = \prod_{1 \leq j < k \leq n} (x_j-x_k).$

The Schur polynomials are defined as the ratio:

$s_{(d_1, d_2, \dots, d_n)} (x_1, x_2, \dots, x_n) = \frac{ a_{(d_1+n-1, d_2+n-2, \dots, d_n+0)} (x_1, x_2, \dots, x_n)} {a_{(n-1, n-2, \dots, 0)} (x_1, x_2, \dots, x_n) }.$

This is a symmetric function because the numerator and denominator are both alternating, and a polynomial since all alternating polynomials are divisible by the Vandermonde determinant.

Read more about this topic: Schur Polynomial

Famous quotes containing the word definition:

“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)

“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)