λ-ring

In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λⁿ on it behaving like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-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 (Lascoux (2003)).

λ-rings were introduced by Grothendieck (1957, 1958, p.148). For more about λ-rings see Atiyah & Tall (1969), Knutson (1973), Hazewinkel (2009) and Yau (2010).

Intuition

If V and W are finite-dimensional vector spaces over a field k, then we can form the direct sum V⊕W, the tensor product V⊗W, and the n-th exterior power of V, Λⁿ(V). All of these are again finite-dimensional vector spaces over k. The same three operations of direct sum, tensor product and exterior power are also available when working with k-linear representations of a finite group, and when working with vector bundles over some topological space, and in more general situations.

λ-rings are designed to abstract the common algebraic properties of these three operations, where we also allow for formal inverses with respect to the direct sum operation. The addition in the ring corresponds to the direct sum, the multiplication in the ring corresponds to the tensor product, and the λ-operations to the exterior powers. For example, the isomorphism

\Lambda^2(V\oplus W)\cong \Lambda^2(V)\oplus\left(\Lambda^1(V)\otimes\Lambda^1(W)\right)\oplus\Lambda^2(W)

corresponds to the formula

\lambda^2(x+y)=\lambda^2(x)+\lambda^1(x)\lambda^1(y)+\lambda^2(y)

valid in all λ-rings, and the isomorphism

\Lambda^1(V\otimes W)\cong \Lambda^1(V)\otimes\Lambda^1(W)

corresponds to the formula

\lambda^1(xy)=\lambda^1(x)\lambda^1(y)

valid in all λ-rings. Analogous but (much) more complicated formulas govern the higher order λ-operators.

Definition

A λ-ring is a commutative ring R together with operations λⁿ:R→R for every non-negative integer n. These operations are assumed to behave like exterior powers of vector spaces, in the sense that they have the same behavior on sums and products that exterior powers have on direct sums and tensor products of vector spaces, and behave in the same way as exterior powers under composition. In more detail, they have the following properties valid for all x, y ∈ R and all n≥0:

λ⁰(x) = 1
λ¹(x) = x
λⁿ(1) = 0 if n ≥ 2
λⁿ(x + y) = Σ_i+j=n λⁱ(x)λ^j(y)
λⁿ(xy) = P_n(λ¹(x), ..., λⁿ(x), λ¹(y), ..., λⁿ(y))
λ^m(λⁿ(x)) = P_m,n(λ¹(x), ..., λ^mn(x))

where P_n and P_m,n are universal polynomials with integer coefficients describing the behavior of exterior powers on tensor products and under composition, that can be described as follows.

Suppose a commutative ring has elements x = x₁ + x₂ + ...,y = y₁ + y₂ + ... and define λⁿ(x) by

$\displaystyle\sum_m \lambda^m(x)t^m=\prod_i (1+tx_i)$

and similarly for y. Informally we think of x and y as vector bundles that are sums of line bundles x_i, y_j, and think of λⁿ(x) as the nth exterior power of x. Then the polynomials P_n and P_m,n are the universal polynomials such that

$\displaystyle\sum_m P_m(\lambda^1(x),\cdots,\lambda^m(x),\lambda^1(y),\cdots,\lambda^m(y))t^m=\prod_{i,j} (1+tx_iy_j)$
$\displaystyle\sum_m P_{m,n}(\lambda^1(x),\cdots,\lambda^{mn}(x))t^m=\prod_{i_1<i_2<\cdots<i_n} (1+tx_{i_1}x_{i_2}\cdots x_{i_n})$ for every integer n≥1.

Some authors call these special λ-rings, and use λ-ring for a more general concept where the conditions on λⁿ(1), λⁿ(xy) and λ^m(λⁿ(x)) are dropped.

Examples

The ring of integers, with the binomial coefficients λⁿ(x) = (x
n) as operations, is a λ-ring.
More generally, any binomial ring becomes a λ-ring if we define the λ-operations to be the binomial coefficients, λⁿ(x) = (x
n). In these λ-rings, all Adams operations are the identity.
The K-theory K(X) of a topological space X is a λ-ring, with the lambda operations induced by taking exterior powers of a vector bundle.
Given a group G, the representation ring R(G) is a λ-ring; the λ-operations are induced by the exterior powers of representations of the group G.
The ring Λ_Z of symmetric functions is a λ-ring. On the integer coefficients the λ-operations are defined by binomial coefficients as above, and if e₁, e₂, ... denote the elementary symmetric functions, we set λⁿ(e₁) = e_n. Using the axioms for the λ-operations, and the fact that the functions e_k generate the ring Λ_Z, this definition can be extended in a unique fashion so as to turn Λ_Z into a λ-ring. In fact it is the universal λ-ring generated by one element.

References

Soulé, C.; Abramovich, Dan; Burnol, J.-F.; Kramer, Jürg (1992). Lectures on Arakelov geometry. Cambridge Studies in Advanced Mathematics 33. Joint work with H. Gillet. Cambridge: Cambridge University Press. ISBN 0-521-47709-3. Zbl 0812.14015.

λ-ring

Intuition

Definition

Examples

References

Further reading