A $\lambda$-ring or lambda ring is a commutative ring together with some operations $\lambda^n$ on it that behave like the exterior powers of vector spaces.
In algebra, a $\lambda$-ring or lambda ring is a commutative ring together with some operations $\lambda^n$ on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural $\lambda$-ring structure. $\lambda$-rings also provide a powerful formalism for studying an action of the symmetric functions on the ring of polynomials, recovering and extending many classical results.
A $\lambda$-ring is a commutative ring $R$ together with operations $\lambda^n : R \to R$ for every non-negative integer $n$. These operations are required to have the following properties valid for all $x, y \in R$ and all $n, m \ge 0$:
- $\lambda^0(x) = 1$
- $\lambda^1(x) = x$
- $\lambda^n(1) = 0$ if $n \ge 2$
- $\lambda^n(x + y) = \sum_{i+j=n} \lambda^i(x) \lambda^j(y)$
- $\lambda^n(xy) = P_n\big(\lambda^1(x), \dots, \lambda^n(x), \lambda^1(y), \dots, \lambda^n(y)\big)$
- $\lambda^n\big(\lambda^m(x)\big) = P_{n,m}\big(\lambda^1(x), \dots, \lambda^{mn}(x)\big)$
where $P_n$ and $P_{n,m}$ are certain universal polynomials with integer coefficients that describe the behavior of exterior powers on tensor products and under composition. These polynomials can be defined as follows.
Let $e_1,\dots,e_{mn}$ be the elementary symmetric polynomials in the variables $X_1,\dots,X_{mn}$. Then $P_{n,m}$ is the unique polynomial in $nm$ variables with integer coefficients such that $P_{n,m}(e_1, \dots, e_{mn})$ is the coefficient of $t^n$ in the expression $$ \prod\limits_{1\le i_1<i_2<\dots <i_m\le mn}\left(1+t X_{i_1} X_{i_2}\dots X_{i_m}\right) $$
(Such a polynomial exists, because the expression is symmetric in the $X_i$ and the elementary symmetric polynomials generate all symmetric polynomials.)
Now let $e1,\dots,e_n$ be the elementary symmetric polynomials in the variables $X_1,\dots,X_n$ and $f_1,\dots,f_n$ be the elementary symmetric polynomials in the variables $Y_1,\dots,Y_n$ . Then $P_n$ is the unique polynomial in $2n$ variables with integer coefficients such that $P_n(e_1,\dots,e_n,f_1,\dots,f_n)$ is the coefficient of $t^n$ in the expression $$ \prod _{i,j=1}^n\left(1+t X_i Y_j\right) $$
The $\lambda$-rings defined above are called "special $\lambda$-rings" by some authors, who use the term "$\lambda$-ring" for a more general concept where the conditions on $\lambda^n(1)$, $\lambda^n(xy)$ and $\lambda^m\big(\lambda^n(x)\big)$ are dropped.
Source: Wikipedia