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)](http://upload.wikimedia.org/math/a/8/1/a817b859e59adc8c4809afaf3ef582a4.png)
![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_3^2](http://upload.wikimedia.org/math/7/e/2/7e209a9f378b764c8a1c42f837578c63.png)
and so forth. Summarizing:
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,
is obviously a symmetric polynomial which is homogeneous of degree four, and we have
Read more about this topic: Schur Polynomial
Famous quotes containing the word example:
“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)