# 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 $$A_{(k)}=A^k_{(k)}.$$

• Should your equation say $p(x_1,\dots,x_{n-1},0)=p(x_1,\dots,x_{i-1},0,x_{i+1},\dots,x_{n})$? – Dan Petersen Apr 15 '15 at 10:35
• No, but it was still wrong, so I fixed it in a way that makes sense. :) – Jakob Oesinghaus Apr 15 '15 at 11:29
• Could you tell us as the Chow ring of what it appears ? – F. C. Apr 24 '15 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. – Jakob Oesinghaus Apr 25 '15 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 :) – darij grinberg Jul 4 '18 at 14:24

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.