Schur Polynomial - Definition

Definition

Schur polynomials correspond to integer partitions. Given a partition

(where each dj 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.

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