> There is also a notion of special lambda-ring. But in most cases by ‘’$\lambda$-ring’‘ is meant ‘’special $\lambda$-ring’’.
symmetric monoidal (∞,1)-category of spectra
A $\lambda$-ring is a commutative ring which is in addition equipped with operations that behave as the operations of forming exterior powers (of vector spaces/representations) in a representation ring. The name derives from the common symbol $\Lambda^n$ for the $n$th exterior power. Hence $\lambda$-rings are one incarnation of the representation theory of the symmetric groups.
Equivalently, it turns out (Wilkerson 82) that a $\lambda$-ring is a commutative ring equipped with an endomorphism that lifts the Frobenius endomorphism after reduction mod $p$ at each prime number $p$. As such, $\lambda$-rings appear in Borger's absolute geometry and are related, in some way, to power operations (see there for more) in stable homotopy theory.
Typically one can form direct sums of representations of some algebraic structure. The decategorification to isomorphism classes of such representations then inherits the structure of a commutative monoid. But nobody likes commutative monoids: we all have an urge to subtract. So, we throw in formal negatives and get an abelian group — the Grothendieck group.
In many situations, we can also take tensor products of representations. Then the Grothendieck group becomes something better than an abelian group. It becomes a ring: the representation ring. Moreover, in many situations we can also take exterior and symmetric powers of representations; indeed, we can often apply any Young diagram to a representation and get a new representation. Then the representation ring becomes something better than a ring: it becomes a $\lambda$-ring.
More generally, the Grothendieck group of a monoidal abelian category is always a ring, called a Grothendieck ring. If we start with a braided monoidal abelian category, this ring is commutative. But if we start with a symmetric monoidal abelian category, we get a $\lambda$-ring.
So, $\lambda$-rings are all about getting the most for your money when you decategorify a symmetric monoidal abelian category — for example the category of representations of a group, or the category of vector bundles on a topological space.
Unsurprisingly, the Grothendieck group of the free symmetric monoidal abelian category on one generator is the free $\lambda$-ring on one generator. This category is very important in representation theory. Objects in this category are called Schur functors, because for obvious reasons they act as functors on any symmetric monoidal abelian category. The irreducible objects in this category are called ‘Young diagrams’. Elements of the free $\lambda$-ring on one generator are called symmetric functions.
A $\lambda$-ring $L$ is a P-ring presented by the polynomial ring $Symm=\mathbb{Z}[h_1,h_1\dots]$ in countably many indeterminates over the integers or, equivalently, $Symm$ is the ring of symmetric functions in countably many variables. This means that (the underlying set valued functor of) $L$ is a copresheaf presented by $Symm$ such that
$L:\CRing \to CRing$ defines an endofunctor on the category of commutative rings.
$L$ gives rise to a comonad on $C Ring$.
A $\lambda$-ring is hence a commutative ring equipped with a co-action of this comonad. As always is the case with monads and comonads this definition can be formulated in terms of an adjunction.
A $\lambda$-structure on a commutative unital ring $R$ is defined to be a sequence of maps $\lambda^n$ for $n\ge 0$ satisfying
$\lambda^0(r)=1$ for all $r\in R$
$\lambda^1=id$
$\lambda^n(1)=0$, for $n\gt 1$
$\lambda^n(r+s)=\sum_{k=0}^n \lambda^k(r)\lambda^{n-k}(s)$, for all $r,s\in R$
$\lambda^n(r s)=P_n(\lambda^1(r),\dots,\lambda^n(r),\lambda^1(s),\dots,\lambda^n(s))$ for all $r,s\in R$
$\lambda^m(\lambda^n(r)):=P_{m,n}(\lambda^1(r),\dots,\lambda^{m n}(r))$, for all $r\in R$
where $P_n$and $P_{m,n}$ are certain (see the reference for their calculation) universal polynomials with integer coefficients. $R$ is in this case called a $\lambda$-ring. Note that the $\lambda^n$ are not required to be morphisms of rings.
A homomorphism of $\lambda$-structures is defined to be a homomorphism of rings commuting with all $\lambda^n$ maps.
There exists a $\lambda$-ring structure on the ring $1+ R[ [t] ]$ of power series with constant term $1$ where
a) addition on $1+R[ [t] ]$ is defined to be multiplication of power series
b) multiplication is defined by
c) the $\lambda$-operations are defined by
Let $\Lambda$ denote the ring of symmetric functions, let $R$ be a $\lambda$-ring.
Then for every $x\in R$ there is a unique homomorphism of $\lambda$-rings
sending $e_1\mapsto x$, $e_n\mapsto \lambda^n(x)$, $p_n\mapsto\psi^n(x)$ where $e_n$ denotes the $n$-th elementary symmetric function and $\psi^n$ denotes the $n$-th Adams operation (explained in the reference).
Equivalently this result asserts that $\Lambda$ is the free $\lambda$-ring in the single variable $e_1$.
This is due to (Hopkinson)
We define $\Phi_x(e_1)=x$, then the assumption on $\Phi_x$ to be a morphism of $\lambda$-rings yields $\Phi_x(e_n)=\Phi_x(\lambda^n(e_1))=\lambda^n(x)$.
(Hazewinkel 1.11, 16.1)
a) The endofunctor of the category of commutative rings
sending a commutative ring to the set of power series with constant term $1$ is representable by the polynomial ring $Symm \coloneqq \mathbb{Z}[h_1, h_2,\dots]$ in an infinity of indeterminates over the integers.
b) There is an adjunction $(forget\, \lambda\dashv \Lambda)$ where $forget\,\lambda: \lambda Ring\to CRing$ is the forgetful functor assigning to a $\lambda$-ring its underlying commutative ring.
The left inverse $\g_{S,A}$ of the natural isomorphism $q_{S,A}:hom(forget\,\lambda,A)\to hom(S,\Lambda(A))$ is given by the ghost component $s_1$.
(see also (Borger 08, section 1.8))
An instructive introduction to the “orthodox”- and preparation for the “heterodox” view (described below) on $\lambda$-rings is Hazewinkel’s survey article on Witt vectors, (Hazewinkel). There is also a reading guide to that article.
There is a second, “heterodox” way to approach $\lambda$-rings with a strong connection to arithmetic discussed in detail in (Borger 08, section 1). An survey is in (Borger 09) where it says in the abstract:
The theory of $\Lambda$-rings, in the sense of Grothendieck’s Riemann–Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring $\mathbf{Z}$ of integers to produce $\Lambda$-algebraic geometry. We show that $\Lambda$-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than $\mathbf{Z}$ and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this in a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.
First some standard notation:
For $p$ a prime number write $\mathbb{F}_p$ for the finite field whose underlying abelian group is the cyclic group $\mathbb{Z}/p\mathbb{Z}$.
For $A$ an $\mathbb{F}_p$-algebra, then the Frobenius endomorphism
is that given by taking each element to its $p$th power
For $p$ a prime number, then a $p$-typical $\Lambda$-ring is
a commutative ring $R$
equipped with an endomorphism $F_A \colon A \to A$
such that under tensor product with $\mathbb{F}_p$ it becomes the Frobenius morphism, def.2:
A big $\Lambda$-ring is a commutative ring equipped with commuting endomorphisms, one for each prime number $p$, such that each of them makes the ring $p$-typical, respectively, as above.
This is def. 1.7 in (Borger 08), formulated for the special case of example 1.15 there (which is stated in terms of Witt vectors) and translated to $\Lambda$-rings in view of prop. 1.10 c) (see the adjunction) there.
> the following originates from revision 19 but needs attention
The $p$-th Adams operation $\psi_p$ is a Frobenius lift. Moreover given any two prime numbers then their Adams operations commute with each other.
The following two theorems are crucial for the “heterodox” point of view. We will see later that in fact we do not need the torsion-freeness assumption.
Let $A$ be an additively torsion-free commutative ring. Let $\{\psi_p\}$ be a commuting family of Frobenius lifts.
Then there is a unique $\lambda$-ring structure on $A$ whose Adams operations are the given Frobenius lifts $\{\psi_p\}$.
A ring morphism $f$ between two $\lambda$-rings is a morphism of $\lambda$-rings (i.e. commuting with the $\lambda$-operations) iff $f$ commutes with the Adams operations.
There is an equivalence between the category of torsion-free $\lambda$-rings and the category of torsion-free commutative rings equipped with commuting Frobenius lifts..
Now we will argue that these statements hold for arbitrary commutative rings.
a) The category $\lambda Ring$ of $\lambda$-rings is monadic and comonadic over the category of $C Ring$ of commutative rings.
b) The category $\lambda Ring_{\neg tor}$ of $\lambda$-rings is monadic and comonadic over the category of $C Ring_{\neg tor}$ of commutative rings.
Let $\i:C Ring_{\neg tor}\hookrightarrow C Ring$ be the inclusion. Let $W^\prime$ denote this comonad on $C Ring_{\neg tor}$. Then
a) $W \coloneqq Lan_i i\circ W^\prime \colon C Ring\to CRing$ is a comonad.
b) The category of coalgebras of $W$ is equivalent to the category of $\lambda$-rings.
c) $W$ is the big-Witt-vectors functor.
The “heterodox” generalizes to arbitrary Dedekind domains with finite residue field.
For instance over $F_p[x]$ (instead of $\mathbb{Z}$), we would look at families of $\psi$-operators indexed by the irreducible monic polynomials $f(x)$, and each $\psi_{f(x)}$ would have to be congruent to the $q$-th power map modulo $f(x)$, where $q$ is the size of $F_p[x]/(f(x))$.
The topological K-theory ring $K(X)$ of any topological space carries the structure of a $\lambda$-ring with operations induced from (skew-)symmetrized tensor products of vector bundles.
This is originally due to Alexander Grothendieck. See for instance Wirthmuller 12, section 11 and see at Adams operations.
The equivariant elliptic cohomology at the Tate curve $Ell_{Tate}(X//G)$ is a $\lambda$-ring (and even an “elliptic $\lambda$-ring”).
The forgetful functor $U \;\colon\; \Lambda Ring \longrightarrow CRing$ from $\Lambda$-rings to commutative rings has
a left adjoint, given by forming the ring $Symm$ of symmetric functions;
a right adjoint given by forming the ring of Witt vectors $W$.
Hence
rings of Witt vectors are the co-free Lambda-rings;
rings of symmetric functions are the free Lambda-rings.
This statement appears in (Hazewinkel 08, p. 87, p. 97, 98). The right adjoint in a more general context is in (Borger 08, prop. 1.10 (c)).
On the level of toposes (etale toposes) over these sites of rings, this statement reappears as an essential geometric morphism from the etale topos of Spec(Z) to that over “F1” in Borger's absolute geometry (Borger 08, exposition in Borger 09).
The $\lambda$-ring structure on topological K-theory goes back to Alexander Grothendieck in the 1960s.
The relation to lifts of Frobenius homomorphisms is due to
Modern accounts include
Michiel Hazewinkel, Formal groups and applications
See also
John Baez, comment.
John R. Hopkinson, Universal polynomials in lambda-rings and the K-theory of the infinite loop space $tmf$, thesis, pdf
Donald Knutson, $\lambda$-Rings and the Representation Theory of the Symmetric Group, Lecture Notes in Mathematics, Vol. 308, Springer, Berlin, 1973.
Donald Yau, LAMBDA-RINGS, World Scientific, 2010.
school/conference in Leiden: Frobenius lifts and lambda rings 5-10. October 2009 featuring
Pierre Cartier: Lambda-rings and Witt vectors
Lars Hesselholt: The de Rham-Witt complex
Alexandru Buium: Arithmetic differential equations
James Borger: Lambda-algebraic geometry
The $\lambda$-ring structure on equivariant elliptic cohomology is due to
Nora Ganter, Stringy power operations in Tate K-theory, Homology, Homotopy, Appl., 2013; arXiv:math/0701565
Nora Ganter, Power operations in orbifold Tate K-theory“; arXiv:1301.2754
Discussion in the context of Borger's absolute geometry is in
James Borger, section 1 of The basic geometry of Witt vectors, I: The affine case (arXiv:0801.1691)
James Borger, Lambda-rings and the field with one element (arXiv/0906.3146)