The stable graded ring of symmetric functions

For integers $N,d\ge 0$, let $A_N^d$ be the degree-$d$ homogeneous part of $\mathbb{Z}[x_1,\dots ,x_N{]}^{S_N}$, using Symmetric polynomials as the invariants of variable permutations and the convention $A_0^0=\mathbb{Z}$ and $A_0^d=0$ for $d>0$. For $M\ge N$, the transition ${\pi }_{M,N}:A_M^d\to A_N^d$ sets $x_{N+1},\dots ,x_M$ equal to zero. These maps compose, so define

An element of ${\mathrm{\Lambda }}^d$ is a compatible sequence of homogeneous degree-$d$ symmetric polynomials, one in each rank. Multiplication of a degree-$a$ sequence and a degree-$b$ sequence is coordinatewise polynomial multiplication and lies in ${\mathrm{\Lambda }}^{a+b}$; compatibility follows because each ${\pi }_{M,N}$ is a ring homomorphism. Extend this product distributively to $\mathrm{\Lambda }$. The unit is the compatible constant sequence $1$, and each element of $\mathrm{\Lambda }$ has only finitely many nonzero homogeneous components.

This direct sum is not the ungraded inverse limit of the rings $\mathbb{Z}[x_1,\dots ,x_N{]}^{S_N}$. For example, the compatible sequence ${\prod }_{i=1}^N(1+x_i)$ belongs to that ungraded inverse limit and has nonzero homogeneous components in every degree, so it is not an element of the direct sum $\mathrm{\Lambda }$. The partition notation and empty partition follow Partitions, English diagrams, and conjugation.