Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

A $\Lambda$-ring structure on a torsion-free ring over $\mathbb Z$ is a commuting family of endomorphisms $\psi_p$ satisfying $\psi_p(x) \equiv x^p$ mod $p$.

One $\Lambda$-ring structure on $\mathbb Z[x]$ is defined by $\psi_p(x)=x^p$.

Another can be defined in terms of the Chebyshev polynomials of the first kind, where $\psi_p(y)=2 T_p(y/2)$. One can view this $\Lambda$-ring as the quotient of the toric $\Lambda$-ring $\mathbb Z[z,z^{-1}]$ where $\psi_p(z)=z^p$ by the $\psi$-equivariant automorphism $z\to z^{-1}$, under the identification $y=z+z^{-1}$.

A result of F.J.-B.J. Clauwens in Commuting polynomials and $\lambda$-ring structures on $\mathbb Z[x]$ shows that these are the only two $\Lambda$-ring structures on $\mathbb Z[x]$ up to isomorphism, but according to the paper Lambda-rings and the field with one element by James Borger, it is not even known whether there are finitely many $\Lambda$-structures on $\mathbb Z[x,y]$ up to isomorphism.

How many non-isomorphic $\Lambda$-ring structures on $\mathbb Z[x,y]$ are known?

All the $\Lambda$-ring structures I know how to construct come from quotients of toric varieties by $\psi$-equivariant group actions.

Are there any $\Lambda$-ring structures on $\mathbb Z[x,y]$ known which are not quotients of toric varieties by group actions? Are there any known which behave very differently from such quotients?

share|improve this question
add comment

1 Answer

This answer contains a list of all the $\Lambda$-ring structures that I have found. If anyone can find any others, they should edit this list.

  • The two $\Lambda$-ring structures on $\mathbb A^1$ give three product $\Lambda$-ring structures on $\mathbb A^2 = \mathbb A^1 \times \mathbb A^1$.

  • One of these product structures, $\psi_p(x)=x^p$ and $\psi_p(y)=y^p$, has a symmetry of order two switching $x$ and $y$. The ring of invariants of this symmetry is another structure.

  • One can look at the ring of invariants of $\mathbb Z[x,x^{-1},y,y^{-1}]$ with $\psi_p(x)=x^p$, $\psi_p(y)=y^p$ under the action of a subgroup of $GL_2(\mathbb Z)$, where $\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)$ sends $x$ to $x^ay^c$ and $y$ to $x^by^d$. If that subgroup is $D_3$, $D_4$, or $D_6$, one gets a $\Lambda$-ring structure on $\mathbb Z[x,y]$. These can also be viewed as the $K$-theory $\Lambda$-rings of the simple Lie groups $SL_3$, $SP_4$, and $G_2$, respectively.

That is a total of $7$ structures.


Here is a proof sketch that they are distinct: These can all be expressed as $\psi_p$-stable subrings of the $\Lambda$-ring $\mathbb Z[x,x^{-1},y,y^{-1}]$ with $\psi_p(x)=x^p$, $\psi_p(y)=y^p$, with the induced $\Lambda$-ring structure. It is easy to check that they are not isomorphic as subrings. Thus it sufficies to recover the embedding into that ring.

Extend $\psi_p$ to $\psi_n$ the obvious way: so that $\psi_n \circ \psi_m = \psi_{nm}$. Then, for any element $\alpha\in \mathbb Z[x,x^{-1},y,y^{-1}]$, the sequence $\psi_n(\alpha)$ is a sum of geometric progressions and so satisfies a finite linear recurrence relation. So if $\alpha$ is a generic element of a $\Lambda$-subring $R$ of dimension $2$, the recurrence relation is defined over $R$, and the roots generate $\mathbb Z[x,y,x^{-1},y,y^{-1}]$. By picking a generic element, we can determine, purely from the $\Lambda$-ring structure on $R$, its embedding, and thereby distinguish the different rings.


EDIT: In fact, the first two constructions suffice to produce infinitely many $\Lambda$-rings! Consider the product ring $\mathbb Z[x,y]$ where $\psi_p(x)=x^p$ and $\psi_p(y)=2T_p(y/2)$. Then the subring $\mathbb Z[x,yx^n]$ is a sub-$\Lambda$ ring for each $n\geq 0$, and none of these are isomorphic. To check that they are non-isomorphic, one can recover $x$ as the unique solution to $\psi_2(x)=x^2$, then invert $x$, then recover $y$ as the unique solution to $\psi_2(y)=y^2-2$ that generates the ring along with $x$ and $x^{-1}$. Thus the embedding of the ring into $\mathbb Z[x,x^{-1},y]$ is unique, so the embedding into $\mathbb Z[x,y]$ is unique, and because they are distinct as subrings, they are distinct.

Similarly, inside $\mathbb Z[x,y]$ where $\psi_p(x)=x^p$ and $\psi_p(y)=y^p$, there is the subring $\mathbb Z[x^{n+1}y^n+x^ny^{n+1},xy]$ which is a sub-$\Lambda$-ring and depends on $n$ for similar reasons.

However these constructions are birationally equivalent to previous defined constructions. So perhaps it is better to consider this up to birational equvialence!

share|improve this answer
    
(1) I think it's better to overcount and explicitly prune, that is, to add an extra instance to construction 2, the same as the $D_4$ invariants, and to add an extra instance to construction 3, which appears in construction 1; (2) you can identify these seven examples with the seven reflection groups in $GL_2(\mathbb Z)$ by saying that the ring of invariants is a localization of the lambda-ring. But that isn't a construction, so you still need all three constructions. –  Ben Wieland Nov 13 '12 at 20:27
    
Yes, that's roughly how I think about them. Indeed, one can view the overlap between constructions 1 and 3 as the K-theory of the semisimple Lie group $SL_2 \times SL_2$. I chose this way for its compactness of presentation, but opinions on which is best can of course differ. –  Will Sawin Nov 14 '12 at 1:17
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.