λ-ring

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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[edit]

If V and W are finite-dimensional vector spaces over a field k, then we can form the direct sum VW, the tensor product VW, 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[edit]

A λ-ring is a commutative ring R together with operations λn:RR 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, yR and all n≥0:

  • λ0(x) = 1
  • λ1(x) = x
  • λn(1) = 0 if n ≥ 2
  • λn(x + y) = Σ i+j=n  λi(xj(y)
  • λn(xy) = Pn1(x), ..., λn(x), λ1(y), ..., λn(y))
  • λmn(x)) = Pm,n1(x), ..., λmn(x))

where Pn and Pm,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 xi, yj, and think of λn(x) as the nth exterior power of x. Then the polynomials Pn and Pm,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 λmn(x)) are dropped.

Examples[edit]

  • 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 ΛZ 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) = en. Using the axioms for the λ-operations, and the fact that the functions ek generate the ring ΛZ, this definition can be extended in a unique fashion so as to turn ΛZ into a λ-ring. In fact it is the universal λ-ring generated by one element.

References[edit]

Further reading[edit]