How to express Witt vectors by their ghost components

Let $x_1,x_2,\cdots$ be infinitely many indeterminates, the ghost components of the Witt vector $x=(x_1,x_2,\cdots)$ is given by \begin{equation*} x^{(n)}:=\sum_{d\mid n}dx_d^{n/d}. \end{equation*}

How to express each $x_n$ as rational polynomial of the ghost components $x^{(n)}$?

Eidt: To be more precise, I wonder if there is a formula like Möbius inversion formula.

Solution 1:

Recall that the Witt vectors of a ring is a functor that is representable by a ring scheme. I will describe the ring scheme below and explain how this will give you a formula for the $x_n$.

Consider $\Lambda,$ the ring of elementary symmetric functions. More concretely, this is the same as those expressions in $\mathbb{Z}[x_1,x_2,\ldots ]$ that are invariant under any permutation of the variables. In this ring you have the Adams operations, these are the expressions $x^{(n)} = x_1^n+x_2^n+x_3^n + \cdots.$ These are what will amount to the ghost components. Let $w_d$ be the symmetric functions determined by the expressions $$x^{(n)} = \sum_{d |n} d w^{n/d}_d.$$

One can then check that $\text{Spec } \Lambda$ can be endowed with a ring scheme structure and that both $x^{(n)} $ and $w_d$ are generating sets of $\Lambda.$ Indeed, these are related by the equality of formal power series $$exp(-\sum_{n\geq 1} \frac{1}{n} x^{(n)} t^n ) = \Pi_{n \geq 1} (1-w_nt^n).$$

Since what I call $w_d$ is precisely the same as what you call $x_d,$ you can by this formula determine them from $x^{(n)}$ recursively. For example, obviously $x^{(1)}=w_1$ and the coefficient in front of $t^2$ on the LHS of the formal power series is given by $-x^{(2)}/2 +(x^{(1)})^{2}/2$ so that $-w_2 = -x^{(2)}/2+(x^{(1)})^{2}/2.$ Negating this, you get precisely an expression for $w_2$ in terms of $x^{i},$ and you can continue this on indefinitely.

This is the easiest formula I know of relating the ghost components and the Adams operations.