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Stream: learning: questions

Topic: Lambda-rings and unary operations

James Deikun (Apr 23 2025 at 05:26):

I'm investigating to what extent left adjoint monads are "unary" and lambda-rings are kind of a paradigmatic case. At [[Lambda-ring]] there are two descriptions, the "orthodox" in terms of lambda operations and the "heterodox" in terms of Adams operations. The lambda operations are "kind of unary", in that they all take one argument, but they are not ring homomorphisms and the relations among them involve complicated polynomials. Those are still unary since they're polynomials in lambdas of a single variable, but they intimately involve the non-unary ring operations.

The Adams operations, on the contrary, are "very unary", in that they both are ring homomorphisms and have relations among themselves that only involve the Adams operations themselves and not the ring operations. This is nice in that the presentation as Adams operations lifts to a left adjoint monad on Set, namely the "actions of the multiplicative monoid of the naturals" monad.

It would be nice if every left adjoint monad had a presentation as nice as the Adams operations. I'm not even sure lambda-rings have a presentation that nice though! The page [[symmetric functions]] describes (without a clear reference or proof) that the power-sum symmetric functions, which correspond to the Adams operations, don't generate $\Lambda [\mathbb Z]$, the ring of symmetric functions over $\mathbb Z$, which should be the free lambda-ring on $\mathbb Z$, and I don't know how to square this with the claim on the other page that the nice Adams operations give a presentation of the theory of lambda-rings.

Can anyone clarify this, or the claim for left adjoint monads in general?

John Baez (Apr 23 2025 at 05:56):

Symmetric functions are certain 'polynomials' in infinitely many variables, invariant under all permutations. But let me see if I can show that the power-sum symmetric polynomials in two variables, namely

$1 + 1, x + y, x^2 + y^2, x^3 + y^3 , \dots$

don't generate the ring of symmetric polynomials in $x$ and $y$ - that is, polynomials in $x$ and $y$ that are invariant under permutations of these two variables. At least, not over $\mathbb{Z}$.

The degenerate case $1 + 1$ is a bit scary, since we have to divide that by $2$ to get $1$, but when we generate a ring we get $1$ for free, so it's not actually a problem.

More of a problem is whether we can get $x y$.

We can get

$(x + y)^2 - (x^2 + y^2) = 2 xy$

but that's not good enough over $\mathbb{Z}$, since we can't divide by 2.

So I think this sort of thing is the reason why

the power-sum symmetric functions, which correspond to the Adams operations, don't generate $\Lambda [\mathbb Z]$, the ring of symmetric functions over $\mathbb Z$....

John Baez (Apr 23 2025 at 05:59):

Over $\mathbb{Q}$ everything should be fine, but of course that's a much weaker statement.