Lambda-ring

Definition

Let $R$ be a commutative ring with $1$, and let $\lambda^n \colon R \to R$ for $n \ge 0$ be maps. Write $\lambda_t(x) = \sum_{n \ge 0} \lambda^n(x) t^n \in R[\![t]\!]$. The pair $(R, (\lambda^n))$ is a lambda-ring if for all $x, y \in R$ and $m, n \ge 1$:

1. $\lambda^0(x) = 1$ and $\lambda^1(x) = x$.
2. $\lambda_t(x + y) = \lambda_t(x) \lambda_t(y)$, that is, $\lambda^n(x + y) = \sum_{i=0}^{n} \lambda^i(x) \lambda^{n-i}(y)$.
3. $\lambda_t(1) = 1 + t$.
4. $\lambda^n(xy) = P_n(\lambda^1(x), \ldots, \lambda^n(x), \lambda^1(y), \ldots, \lambda^n(y))$.
5. $\lambda^m(\lambda^n(x)) = P_{m,n}(\lambda^1(x), \ldots, \lambda^{mn}(x))$.

Here $P_n$ and $P_{m,n}$ are the universal polynomials with integer coefficients defined as follows. Take variables $\xi_1, \ldots, \xi_N$ and $\eta_1, \ldots, \eta_N$ with $N$ large, and let $e_k$ be the elementary symmetric polynomials. Then $P_n(e_1(\xi), \ldots, e_n(\xi), e_1(\eta), \ldots, e_n(\eta))$ is the coefficient of $t^n$ in $\prod_{i,j} (1 + \xi_i \eta_j t)$. And $P_{m,n}(e_1(\xi), \ldots, e_{mn}(\xi))$ is the coefficient of $t^m$ in $\prod_{i_1 < \cdots < i_n} (1 + \xi_{i_1} \cdots \xi_{i_n} t)$.

A ring with maps satisfying only clauses 1 and 2 is a pre-lambda-ring.