Time for another gemstone from symmetric function theory! (I am studying for my Ph.D. qualifying exam at the moment, and as a consequence, the next several posts will feature yet more gemstones from symmetric function theory. You can refer back to this post for the basic definitions.)
Start with a polynomial that factors as
The coefficients of are symmetric functions in – in fact, they are, up to sign, the elementary symmetric functions in .
In particular, if denotes , then
(These coefficients are sometimes referred to as Vieta’s Formulas.)
Since for all , we can actually turn the equation above into a symmetric function identity by plugging in :
and then summing these equations:
…and we have stumbled across another important basis of the symmetric functions, the power sum symmetric functions. Defining , every symmetric function in can be uniquely expressed as a linear combination of products of these `s (we write .)
So, we have
This is called the th Newton-Girard identity, and it gives us a recursive way of expressing the `s in terms of the `s.
Well, almost. We have so far only shown that this identity holds when dealing with variables. However, we can plug in any number of zeros for the `s to see that the identity also holds for any number of variables . And, if we had more than variables, we can compare coefficients of each monomial individually, which only can involve at most of the variables at a time since the equation is homogeneous of degree . Setting the rest of the variables equal to zero for each such monomial will do the trick.
So now we have it! The Newton-Girard identities allow us to recursively solve for in terms of the `s. Wikipedia does this nicely and explains the computation, and the result is:
For instance, this gives us , which is true:
The Wikipedia page derives a similar identity for expressing the `s in terms of the `s. It also does the same for expressing the complete homogeneous symmetric functions in terms of the `s and vice versa. However, it does not explicitly express the `s in terms of the `s or vice versa. In the name of completeness of the Internet, let’s treat these here.
Fix some number of variables . For any , define to be the sum of all monomials of degree in . This is clearly symmetric, and we define
for any partition , as we did for the elementary symmetric functions last week. The `s, called the complete homogeneous symmetric functions, form a basis for the space of symmetric functions.
It’s a fun exercise to derive the following generating function identities for and :
(The first requires expanding out each factor as a geometric series, and then comparing coefficients. Try it!)
From these, we notice that , and by multiplying the generating functions together and comparing coefficients, we find the identities
Just as before, this gives us a recursion for in terms of the `s. With a bit of straightforward algebra, involving Cramer’s Rule, we can solve for :
We can also use the same equations to solve for in terms of :
I find these two formulas to be more aesthetically appealing than the standard Newton-Gerard formulas between the `s and `s, since they lack the pesky integer coefficients that appear in the first column of the matrix in the -to- case. While perhaps not as mainstream, they are gemstones in their own right, and deserve a day to shine.