λ-ring

In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λⁿ on it that behave 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).

Motivation

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, 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. (These formal inverses also appear in Grothendieck groups, which is why the underlying additive groups of most λ-rings are Grothendieck groups.) 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.

Motivation with Vector Bundles

If we have a short exact sequence of vector bundles over a smooth scheme $X$

$0\to {\mathcal {E}}''\to {\mathcal {E}}\to {\mathcal {E}}'\to 0,$

then locally, for a small enough open neighborhood $U$ we have the isomorphism

\bigwedge ^{n}{\mathcal {E}}|_{U}\cong \bigoplus _{i+j=n}\bigwedge ^{i}{\mathcal {E}}'|_{U}\otimes \bigwedge ^{j}{\mathcal {E}}''|_{U}

Now, in the Grothendieck group $K(X)$ of $X$ (which is actually a ring), we get this local equation globally for free, from the defining equivalence relations. So

{\begin{aligned}\left[\bigwedge ^{n}{\mathcal {E}}\right]&=\left[\bigoplus _{i+j=n}\bigwedge ^{i}{\mathcal {E}}'\otimes \bigwedge ^{j}{\mathcal {E}}''\right]\\&=\sum _{i+j=n}\left[\bigwedge ^{i}{\mathcal {E}}'\right]\cdot \left[\bigwedge ^{j}{\mathcal {E}}''\right]\end{aligned}}

demonstrating the basic relation in a λ-ring,^[1] that

\lambda ^{n}(x+y)=\sum _{i+j=n}\lambda ^{i}(x)\lambda ^{j}(y).

Definition

A λ-ring is a commutative ring R together with operations λⁿ : R → 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 ≥ 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))
λⁿ(λ^m(x)) = P_n,m(λ¹(x), ..., λ^mn(x))

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 e₁, ..., e_mn be the elementary symmetric polynomials in the variables X₁, ..., X_mn. Then P_n,m is the unique polynomial in nm variables with integer coefficients such that P_n,m(e₁, ..., e_mn) is the coefficient of tⁿ in the expression

\prod _{1\leq i_{1}<i_{2}<\cdots <i_{m}\leq mn}(1+tX_{i_{1}}X_{i_{2}}\cdots X_{i_{m}})

(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, ..., e_n be the elementary symmetric polynomials in the variables X₁, ..., X_n and f₁, ..., f_n be the elementary symmetric polynomials in the variables Y₁, ..., Y_n. Then P_n is the unique polynomial in 2n variables with integer coefficients such that P_n(e₁, ..., e_n, f₁, ..., f_n) is the coefficient of tⁿ in the expression

\prod _{i,j=1}^{n}(1+tX_{i}Y_{j})

Variations

The λ-rings defined above are called "special λ-rings" by some authors, who use the term "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 Z of integers, with the binomial coefficients $\lambda ^{n}(x)={x \choose n}$ as operations (which are also defined for negative x) is a λ-ring. In fact, this is the only λ-structure on Z. This example is closely related to the case of finite-dimensional vector spaces mentioned in the Motivation section above, identifying each vector space with its dimension and remembering that $\dim(\Lambda ^{n}(k^{x}))={x \choose n}$ .
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 and a base field k, the representation ring R(G) is a λ-ring; the λ-operations are induced by the exterior powers of k-linear 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 are algebraically independent and 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, this is the free λ-ring on one generator, the generator being e₁. (Yau (2010, p.14)).

Further properties and definitions

Every λ-ring has characteristic 0 and contains the λ-ring Z as a λ-subring.

Many notions of commutative algebra can be extended to λ-rings. For example, a λ-homomorphism between λ-rings R and S is a ring homomorphism f : R → S such that f(λⁿ(x)) = λⁿ(f(x)) for all x in R and all n ≥ 0. A λ-ideal in the λ-ring R is an ideal I in R such that λⁿ(x) ϵ I for all x in R and all n ≥ 1.

If x is an element of a λ-ring and m a non-negative integer such that λ^m(x) ≠ 0 and λⁿ(x) = 0 for all n > m, we write dim(x) = m and call the element x finite-dimensional. Not all elements need to be finite-dimensional. We have dim(x+y) ≤ dim(x) + dim(y) and the product of 1-dimensional elements is 1-dimensional.

References

^ Pieter Belmans (23 October 2014). "Three filtrations on the grothendieck ring of a scheme".

Atiyah, M. F.; Tall, D. O. (1969), "Group representations, λ-rings and the J-homomorphism.", Topology, 8: 253–297, doi:10.1016/0040-9383(69)90015-9, MR 0244387
Expo 0 and V of Berthelot, Pierre; Alexandre Grothendieck; Luc Illusie, eds. (1971). Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics 225) (in French). Berlin; New York: Springer-Verlag. xii+700. doi:10.1007/BFb0066283. ISBN 978-3-540-05647-8. MR 0354655.
Grothendieck, Alexander (1957), "Special λ-rings", Unpublished
Grothendieck, Alexander (1958), "La théorie des classes de Chern", Bull. Soc. Math. France, 86: 137–154, MR 0116023
Hazewinkel, Michiel (2009), "Witt vectors. I.", Handbook of algebra. Vol. 6, Amsterdam: Elsevier/North-Holland, pp. 319–472, arXiv:0804.3888, doi:10.1016/S1570-7954(08)00207-6, ISBN 978-0-444-53257-2, MR 2553661
Knutson, Donald (1973), λ-rings and the representation theory of the symmetric group, Lecture Notes in Mathematics, vol. 308, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0069217, MR 0364425
Lascoux, Alain (2003), Symmetric functions and combinatorial operators on polynomials (PDF), CBMS Reg. Conf. Ser. in Math. 99, American Mathematical Society
Soulé, C.; Abramovich, Dan; Burnol, J.-F.; Kramer, Jürg (1992). Lectures on Arakelov geometry. Cambridge Studies in Advanced Mathematics. Vol. 33. Joint work with H. Gillet. Cambridge: Cambridge University Press. ISBN 0-521-47709-3. Zbl 0812.14015.
Yau, Donald (2010), Lambda-rings, Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., doi:10.1142/7664, ISBN 978-981-4299-09-1, MR 2649360

[1] Pieter Belmans (23 October 2014). "Three filtrations on the grothendieck ring of a scheme".

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