# Identify ring of polynomials symmetric under forgetting variables

I came across the following ring $A$, which appears as a Chow ring. I am wondering if it has been studied before; in particular, I am looking for a reference where this object might have been described.

The graded ring $A^n \subset \mathbb{Z}[x_1,\dots,x_n]$ is the subring consisting of polynomials $p$ such that $$p(x_1,\dots,x_{n-1},0)=p(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1})$$ for all $i\geq 1$. For $m\geq n$, there is a surjective homomorphism $A^m \to A^n$ given by setting the last $m-n$ variables to $0$. This is an isomorphism in degrees smaller than $n$. We define $$A = \varprojlim A^n$$ in the category of graded rings. In particular, the degree $k$ part of $A$ is $$A_{(k)}=A^k_{(k)}.$$

• Could you tell us as the Chow ring of what it appears ? Apr 24, 2015 at 19:53
• This is the Chow ring of the stack of expansions, or equivalently the moduli stack of aligned log structures, as considered e.g. in arXiv:1110.2976. Apr 25, 2015 at 20:16
• So apparently this has led to arXiv:1806.10700. Looks like it might finally get algebraic geometers to care about quasisymmetric functions :) Jul 4, 2018 at 14:24
• @darijgrinberg So far, it has made one algebraic geometer care about them. We will see :). Jul 5, 2018 at 8:50

The ring $A^n$ is the set of polynomials $f\in\mathbb{Z}[x_1,\ldots,x_n]$ such that the coefficient of $x_{i_1}^{a_1}\ldots x_{i_k}^{a_k}$ is equal to the coefficient of $x_{j_1}^{a_1}\ldots x_{j_k}^{a_k}$ whenever $i_1<\ldots<i_k$ and $j_1<\ldots<j_k$. Polynomials satisfying this condition are called quasi-symmetric, and $A$ is the ring of quasi-symmetric functions.