Questions tagged [lambda-rings]

For questions about $\lambda$-rings, which are commutative rings with operations which mimic the behavior of exterior powers of vector spaces.

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$\lambda$-ring endomorphisms of ${\mathbb Z}[x]$

$\DeclareMathOperator\SL{SL}$As explained in this question , there are two $\lambda$-ring structures on ${\mathbb Z}[x]$. In layman's terms, both come from a realization of ${\mathbb Z}[x]$ as the ...
Bugs Bunny's user avatar
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4 votes
0 answers
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What are the applications of $\lambda$-rings to class field theory?

In the book Lambda Rings by Yau, he mentions several areas where $\lambda$-rings can be applied, but he doesn't go into much details. He even includes class field theory in the list, mentioning "...
Lukas Heger's user avatar
9 votes
1 answer
391 views

Linear recurrence relation for symmetric powers in the Burnside ring

Let $G$ be a finite group and $B(G)$ be its Burnside ring, i.e. formal sums of isomorphism classes of finite $G$-sets with addition given by disjoint union and multiplication given by Cartesian ...
Aleksei Piskunov's user avatar
7 votes
0 answers
125 views

How are symmetric functions related to Koszul duality?

Staying within the world of linear algebra, we have the following two "dualities" between exterior powers and symmetric powers. The first is that of Kozsul duality, so these two graded ...
Chris H's user avatar
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6 votes
1 answer
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When and why are Adams operations "non-negative"?

We can think of the unary operations in a lambda-ring as integer linear combinations of Young diagrams; for example the operation $\lambda^n$ corresponds to the Young diagram with $n$ rows and one ...
John Baez's user avatar
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6 votes
0 answers
217 views

Adams operation on Q-construction of fields

Let $F$ be a field that we want to compute its rational algebraic $K$-theory using the Quillen's $Q$-construction. Let $QF$ be the $Q$ construction of the category of finite dimensional vector spaces ...
user127776's user avatar
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3 votes
0 answers
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Adams operation on the rational homology

The Adams operation acts on the algebraic $K$-theory of $R$ but the action as far as I know doesn't come from a endo-functor on the category of projective modules over $R$. For the $K_0$ there is an ...
user127776's user avatar
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13 votes
2 answers
769 views

Categories which are both monadic and comonadic over another category

I heard a professor say that $\lambda$-rings are both monadic and comonadic over commutative rings. Remark 2.11(a) on the nlab page says the same. What does it mean, intuitively, that a category is ...
Arrow's user avatar
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13 votes
1 answer
826 views

Is an ordinary scheme in Borger's Absolute Geometry the same as a "scheme over 𝔽₁" with a map to Spec(ℤ)?

$\newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}_1} \newcommand{\spec}{\operatorname{Spec}}$If I understand correctly, in Borger's paper $\Lambda$-rings and the field with one element about the ...
KotelKanim's user avatar
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3 votes
1 answer
654 views

Witt vectors and maps of $\lambda$-rings

Consider the ring $W(\mathbb{F}_p)$ of big Witt vectors of $\mathbb{F}_p$. This has a natural structure of a $\lambda$-ring (in the strong sense) since rings of big Witt vectors always do. $\mathbb{Z}...
Nate's user avatar
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8 votes
2 answers
538 views

How do I find coefficients of a product expansion

Any power series $f(t) = 1 + t \mathbb{Z}[[t]]$ can be uniquely expanded in the following two ways: $$1 + \sum_{i=1}^\infty f_i t^i = \prod_{i=1}^\infty (1-t^i)^{-n_i}$$ Here, the $f_i$ and $n_i$ ...
Vivek Shende's user avatar
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38 votes
3 answers
2k views

Lambda-operations on stable homotopy groups of spheres

The Barratt-Quillen-Priddy theorem says in one interpretation that there is a weak equivalence of spectra $K(FinSet) \simeq \mathbb{S}^0$. In other words K-theory groups of finite sets are the stable ...
skupers's user avatar
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8 votes
5 answers
1k views

$\lambda$-ring structure defined for a graded ring in Fulton–Lang's book

Given a commutative ring $A$ with unity, Grothendieck used universal polynomials to define a special $\lambda$-ring structure on $\Lambda(A):=1+t\:A[[t]]$. Suppose $A$ is graded, say $A=\bigoplus_{i=0}...
Mahdi Majidi-Zolbanin's user avatar
5 votes
1 answer
360 views

Center of the category of special $\lambda$-rings

Recall that the center $\mathrm{Z}(C)$ of a category $C$ is the monoid of endomorphisms of $\mathrm{id}_C$. Thus $\eta \in \mathrm{Z}(C)$ is given by a familiy of endomorphisms $\eta_x : x \to x$, ...
Martin Brandenburg's user avatar
5 votes
3 answers
591 views

An isomorphism of 2-Schur modules

This is the little brother of question 68071: elementary, simple-looking and probably much easier to answer. Of course, it is just a small part of question 68071, as anybody with $\lambda$-rings ...
darij grinberg's user avatar
5 votes
1 answer
763 views

K-Theory as a special $\lambda$-ring

I wonder if there is a nice and short proof that the $K$-theory of a topological space is a special $\lambda$-ring without invoking the splitting principle and alike. Is it possible to show directly ...
Martin Brandenburg's user avatar
3 votes
1 answer
609 views

Left Adjoint to the Forgetful Functor on $\lambda$-rings?

The forgetful functor from the category of $\lambda$-rings to that of rings has a right adjoint in the form of the universal $\lambda$ functor $\Lambda$, which is isomorphic to the big Witt vectors ...
Joe Johnson's user avatar
11 votes
1 answer
1k views

When are representation rings special lambda-rings? (variations of an old question)

Status: Questions 2 and 4 answered in the negative. Questions 1 and 3 ARE STILL UNANSWERED, despite previous claims. On the third page of Wolfang K. Seiler's paper "lambda-rings and Adams operations ...
darij grinberg's user avatar
9 votes
1 answer
992 views

Is the Burnside ring a lambda-ring? + conjecture in Knutson p. 113

Warning: I'll be using the "pre-$\lambda$-ring" and "$\lambda$-ring" nomenclature, as opposed to the "$\lambda$-ring" and "special $\lambda$-ring" one (although I just used the latter a few days ago ...
darij grinberg's user avatar
8 votes
2 answers
1k views

Is every Adams ring morphism a lambda-ring morphism?

A lambda-ring $R$ is called "special" if it satisfies the $\lambda^i\left(xy\right)=...$ and $\lambda^i\left(\lambda^j\left(x\right)\right)=...$ relations, or, equivalently, if the map $\lambda_T:R\to\...
darij grinberg's user avatar