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.
20
questions
9
votes
1
answer
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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 ...
4
votes
0
answers
151
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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 "...
9
votes
1
answer
394
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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 ...
7
votes
0
answers
126
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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 ...
6
votes
1
answer
325
views
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 ...
6
votes
0
answers
217
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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 ...
3
votes
0
answers
108
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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 ...
13
votes
2
answers
772
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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 ...
13
votes
1
answer
829
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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 ...
3
votes
1
answer
655
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}...
8
votes
2
answers
539
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 $n_i$ ...
38
votes
3
answers
2k
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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 ...
8
votes
5
answers
1k
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$\lambda$-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}...
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 $\eta_x : x \to x$, ...
5
votes
3
answers
608
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 ...
5
votes
1
answer
766
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 ...
3
votes
1
answer
610
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 ...
11
votes
1
answer
1k
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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 ...
9
votes
1
answer
996
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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 ...
8
votes
2
answers
1k
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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\...