So far, we've considered geometric objects. Let's also have an example of something that isn't geometric. Let \(f\) be a polynomial in some number of variables. For now, we'll stick with 3 variables, \(x, y\), and \(z\). We say that \(f\) is a symmetric polynomial if every way of switching around (ie, permuting) the variables leaves \(f\) the same.
For example, the polynomial \(f(x,y,z)=x+y+z\) is symmetric: switching the \(x\) and the \(z\), for example, gives \(z+y+x\), which is the same as \(f\). As a more complicated example, you can check that \(g(x,y,z)=x^2y+x^2z + y^2x + y^2z +z^2x + z^2y\) is also symmetric.
On the other hand, \(h(x,y,z)=x^3+y^3+z\) is not symmetric, since switching \(x\) and \(z\) produces \(z^3+y^3+x\), which is not equal to \(h\). This polynomial does have some symmetry, since switching \(x\) and \(y\) leaves \(h\) the same, but we save the name 'symmetric polynomial' for the fully symmetric polynomials.
Let \(f\) be a symmetric polynomial with \(n\) variables. how many symmetries does \(f\) have?
If you haven't tried a problem like this before - working in \(n\) variables - it is extremely important to get some practice. Try writing down some different symmetric polynomials with small numbers of variables. Is there a formula that describes the the number of symmetries in terms of the number of variables?
Symmetric polynomials are really interesting things, and we'll see them again when we talk about rings and vector spaces!
- Tom Denton (Fields Institute/York University in Toronto)