All Questions
Tagged with f-1 or field-with-one-element
73 questions
9
votes
0
answers
144
views
Ringel's interpretation of quantum groups as Hall algebras at $q=1$
Let $Q$ be a finite-type quiver and let $\mathfrak{g}$ be the semisimple Lie algebra associated with the corresponding simply-laced Dynkin diagram. Let $U_v^+(\mathfrak{g})$ be the positive part of ...
- 3,446
8
votes
1
answer
1k
views
Why is $\operatorname{Spec}(\mathbb Z)$ supposed to lie over $\operatorname{Spec}(\mathbb F_1)$ rather than the other way around?
$\DeclareMathOperator\Spec{Spec}$I understand that one major motivation for the field with one element is supposed to be that there should be a map $\Spec(\mathbb Z) \to \Spec(\mathbb F_1)$, which has ...
- 63.9k
4
votes
0
answers
283
views
Intermediate arithmetic results in F_1 geometry
Much is made of the search for a proof of the Riemann hypothesis via the field with one element. Are there any lesser classic arithmetic results that have been proved with F_1 geometry, such as the ...
- 447
5
votes
0
answers
234
views
Lax monoidal structure on the right Kan extension of a partially monoidal Γ-set
First some preliminaries. Let me write $Fin_\ast$ for the skeleton of the category of finite pointed sets and pointed maps between them on the objects $n_+=\{0,1,...,n\}$, where $0$ is the base point (...
- 10.4k
11
votes
1
answer
948
views
Representations of finite groups over the "field with one element"
Have there been any attempts to extend the "F_un" analogy to the representation theory of finite groups?
If I might be allowed some speculation:
If combinatorics can be regarded as analagous ...
- 1,433
2
votes
0
answers
180
views
Is there a homotopical analogue of short exact sequence?
For $R$-modules for a commutative ring $R$, submodules and quotients are put on roughly the same footing; the kernel of a quotient is an injection into the source, and the cokernel of this injection ...
- 1,252
7
votes
0
answers
295
views
Connections between Borger's absolute geometry and Connes' and Consani's $\Gamma$-spaces
As the idea of an absolute geometry over the field with one element $\mathbb{F}_1$ becomes more clear, two approaches seem to have crystallized, being based on different assumptions and going into ...
6
votes
0
answers
469
views
Étale cohomology of the field with one element
In the function field - number field analogy, some expect progress on RH to come from reproducing various aspects of the Grothendieck program in a way where $\mathbb{Z}$ could be treated as a function ...
user30211
15
votes
0
answers
534
views
Does the $\mathbb{F}_1$ point of view lead to any testable predictions?
In number theory we can informally consider number rings as curves over something like a field with one element. For example it is mentioned here by Kedlaya.
The question is does this perspective lead ...
- 441
5
votes
0
answers
536
views
What do we know about $\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z}$ and the spectral DM Stack $\mathrm{Spét}(\mathbb{Z}\otimes_{\mathbb{S}}\mathbb{Z})$?
These days I've been trying to wrap my head around the current proposed approaches to algebraic geometry over the elusive "field with one element", one of whose main objects of interest is ...
- 11.8k
2
votes
0
answers
199
views
Does the category of rings embed fully faithfully into the category of $\mathbb{F}_{1}$-algebras?
The idea of a theory of algebraic geometry over the "field with one element" $\mathbb{F}_{1}$ is to give a fully faithfully embedding of categories
$$\mathsf{Sch}_{\mathbb{Z}}\hookrightarrow\...
- 11.8k
41
votes
1
answer
3k
views
Connes–Consani's absolute geometry and Lurie's spectral algebraic geometry
Alain Connes and Caterina Consani seem to be currently working on "absolute algebraic geometry", which is a kind of "algebraic geometry over the sphere spectrum" (https://arxiv.org/...
- 511
7
votes
0
answers
181
views
In what sense do the real and complex places correspond to setting q equal to 1 or -1?
It often happens that if we have a scheme $X/\mathbb Z$ (or an open subset thereof) and we denote by $p(q) = X(\mathbb F_q)$, then $p(1)$ and $p(-1)$ compute the euler characteristic of $X(\mathbb C)$ ...
- 7,736
35
votes
2
answers
2k
views
Durov approach to Arakelov geometry and $\mathbb{F}_1$
Durov's thesis on algebraic geometry over generalized rings looks extremely intriguing: it promises to unify scheme based and Arakelov geometry, even in singular cases, as well as including geometry ...
- 14.7k
11
votes
0
answers
401
views
The term "absolute geometry"
My question concerns the so-called absolute geometry over the "field with one element" F_1 or over the spectrum $\mathrm{Spec}(F_1)$, cf. https://ncatlab.org/nlab/show/Borger%27s+absolute+geometry. I ...
- 1,338
5
votes
1
answer
247
views
Generating function for lattice paths making aribitrary (i,j)-up-right move in one step and fitting rectangular (m,n)?
There is the following beautiful formula (see Qiaochu Yuan excellent blog):
$$ \sum_{\lambda \in Young~diagrams~fitting~rectangle~m~n} q^{Box~count(="area~under~the~curve")~of~\lambda} = \binom{n+m}{...
- 24.8k
13
votes
2
answers
641
views
$q$ as a prime power and a root of unity
The number of points on the $(n-1)$-dimensional projective space $P^{n-1}(\mathbb{F}_q)$ over a finite field $\mathbb{F}_q$ is the $q$-integer
$$[n]_q := \frac{q^n-1}{q-1}.$$
In analogy, the number of ...
- 1,430
15
votes
1
answer
747
views
Schur-Weyl duality and q-symmetric functions
Disclaimer: I'm far from an expert on any of the topics of this question. I apologize in advance for any horrible mistakes and/or inaccuracies I have made and I hope that the spirit of the question ...
- 7,789
24
votes
2
answers
1k
views
What is $\mathbb{Q}_1$, the "field of $1$-adic numbers"?
(Disclaimer: I'm totally ignorant about $\mathbb{F}_1$ theory)
There are now (several) working definitions of the "field with one element" $\mathbb{F}_1$ (not literally a field, of course), and ...
- 23.3k
7
votes
1
answer
390
views
Combinatorial/probabilistic statements having $F_{\text{un}}$/$F_q$ geometric interpetation
$\newcommand{\Fun}{F_\text{un}}$There was lots of "Fun with $\Fun$" (field with one element) in recent years.
One of the points is that it provides bridge between geometrical and ...
- 24.8k
7
votes
1
answer
434
views
Can one divide algebraic manifolds ? Make sense: $Gr(2,n)/ Gr(2,n+m) = P^{n-1}/P^{n+m-1} P^{n-2}/P^{n+m-2}$
Let's start from a little bit far.
Basic probability theory - chain rule reads:
$$ P(AB) = P(A)P(B|A)$$
Example: consider n+m balls, where n - white balls, m - black balls,
consider A - first ...
- 24.8k
14
votes
1
answer
801
views
Is there a lift of the q-Vandermonde identity to some geometric (motivic) identity for Grassmannians over $F_q$?
The q-Vandermonde identity reads:
$$ \binom{m + n}{k}_{\!\!q} =\sum_{j} \binom{m}{k - j}_{\!\!q} \binom{n}{j}_{\!\!q} q^{j(m-k+j)} $$
The q-binomial coefficients:
$$ \binom{ a }{ b}_{\!\!q} $$
...
- 24.8k
10
votes
0
answers
343
views
What are the analogs of a Levi/Parabolic/Borel/Bruhat over the field with 1 element?
This is inevitably an imprecise question, but there are already several questions like this on the site so I thought i'd try anyway.
If I understand correctly, for any reductive algebraic group $G$ ...
- 7,789
25
votes
1
answer
1k
views
How is Borger's approach to $\mathbb{F_{1}}$ related to previous approaches (e.g. Deitmar's)?
The "traditional" approach to the so-called "field with one element" $\mathbb{F}_{1}$ is by using monoids, or, to put it in another way, by forgetting the additive structure of ...
- 3,309
15
votes
0
answers
2k
views
Inter-Universal Teichmuller Theory and the Field with One Element
The idea of the "field with one element", or $\mathbb{F}_{1}$, is supposed to allow us to do for number fields what we can do for function fields. Hence this idea often comes up regarding problems ...
- 3,309
5
votes
2
answers
292
views
Field with one element look at counting index-$n$ subgroups in terms of Homs to $S_n$, generalization to $F_{1^k}$?
Main idea shortly: As we discussed recently MO272045, there is beautiful fomula which
counts index-n subgroups in terms of homomorphisms to $S_n$.
Let me give "field with one element" interpretation ...
- 24.8k
9
votes
3
answers
941
views
Buildings, projective geometry - what led Tits to think of "the field with one element"?
The mysterious object "field with one element" seems to appear first in J. Tits papers on buildings. It is mentioned in almost any text on $\mathbb{F}_1$.
However, I have never seen any exposition of ...
- 24.8k
26
votes
1
answer
816
views
What are the points of simple algebraic groups over extensions of $\mathbb{F}_1$?
The "field with one element" $\mathbb{F}_1$ is, of course, a very speculative object. Nevertheless, some things about it seem to be generally agreed, even if the theory underpinning them is not; in ...
- 32.4k
33
votes
2
answers
1k
views
Analogies supporting heuristic: Weyl groups = algebraic groups over field with one element?
There is well-known heuristic that Weyl groups are reductive algebraic groups over "field with one element".
Probably the best known analogy supporting that heuristic is the limit $q\to1$ ...
- 24.8k
13
votes
0
answers
740
views
Bijection between conjugacy classes and irreducible representation of Weyl group = Langlands correspondence over "field with one element"
Context: The number of conjugacy classes equals to the number of irreducuble representations (over C) for any finite group.
Moreover for the symmetric group there is well-known "natural bijection" ...
- 24.8k
11
votes
1
answer
619
views
Gauss, Jacobi, Kloosterman sums and representation theory in the $\mathbb F_1$-world
This question is inspired by Why are Bessel function and Kloosterman sum similar? - it developed in me desire to understand Kloosterman sums better.
There seems to be common knowledge that Gauss, ...
- 18.4k
3
votes
0
answers
209
views
Can MacLane's notion of universality inform $\mathbb{F}_1$?
MacLane (1939) calls a field $F$ universal if every other field $F'$ of the same cardinality and characteristic as $F$ is a subfield of $F$. He then exhibits an example, viz. a field of generalized ...
- 15.4k
16
votes
1
answer
1k
views
What "should" be the absolute galois group of a field with one element
As far as I know there is many "suggestions" of what should be a "field with one element" $\mathbf{F}_{1}$.
My question is the following:
How we should think or what should be the "absolute Galois ...
- 771
5
votes
2
answers
236
views
Descent of flatness from algebras to monoids
Consider a morphism of commutative monoids $u\colon M\rightarrow N$. We say that $u$ is flat, if the tensor product functor $\bullet\otimes_MN$ from the category of $M$-modules to the category of $N$-...
- 6,700
13
votes
1
answer
868
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 ...
- 2,300
1
vote
0
answers
709
views
What is the significance of the $-1$-simplex?
The number of $k$-simplex elements in an $n$-simplex is counted by the binomial coefficient $\binom{n+1}{k+1}$. For example, the $3$-simplex is the tetrahedron, which has the following elements: $4$ ...
6
votes
0
answers
342
views
What is known about the $q$-analogue of the simplex?
I am interested in the field with one element. I am thus interested in combinatorial interpretations of the Gaussian binomial coefficients. Richard Stanley's "Enumerative combinatorics" mentions ...
25
votes
1
answer
2k
views
A geometric theory of Blueprints? (Algebras over the field with one element)
In my attempt to tackle the various approaches of defining algebraic geometry over $\mathbb F_1$, I was just reading through Lorscheid's paper The geometry of blueprints. I certainly like the idea a ...
- 2,303
6
votes
0
answers
294
views
Laurent and power series over the field with one element?
Question. Is there a suitable notion of the Laurent series ring $\mathbb{F}_1((t))$ and power series ring $\mathbb{F}_1[[t]]$ in some framework for the field with one element $\mathbb{F}_1$?
For ...
user1437
3
votes
1
answer
258
views
Are binomial coefficients $F_1$ analogs of $q$-binomial coefficients?
This is a mostly philosophical question. Is it fair to think of usual binomial coefficients and their identities as an $F_1$ case of $q$-binomial coefficients and identities? Here $F_1$ is the field ...
- 5,186
23
votes
1
answer
949
views
Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$?
As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a ...
- 17.4k
15
votes
0
answers
448
views
Connes' idea to use the hyperfinite $III_1$ factor for the archimedian place of Spec(Z)?
I recently gave a talk, where I talked about the tensor category
of (all, not just finite index) bimodules over the hyperfinite $III_1$ factor.
Vaughan Jones, who was in the audience, later told me ...
- 43.2k
8
votes
0
answers
508
views
The logarithm over $\mathbb F_1$
In 'Cyclotomy and analytic geometry over F1', Manin proposes a version of the notion of `analytic function' over the 'field with one element $\mathbb F_1$'.
Question 1: can somebody explain or give ...
- 838
3
votes
0
answers
261
views
Categorizing epimorphisms in $\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)$
This is a follow up to this question of mine. The setup: Let $\mathcal{B}$ be an $\mathbb{F}_1$-linear category (Deitmar uses the term Belian); that is, $\mathcal{B}$ is pointed; balanced; contains ...
- 1,349
11
votes
1
answer
853
views
Is $Lex(\mathcal B,\mathsf{Set}_*)$ an $\mathbb F_1$-linear category?
Following Anton Deitmar, let $\mathcal B$ be an "$\mathbb F_1$-linear category" (Deitmar uses the term "Belian"); i.e., $\mathcal B$ is balanced, pointed, contains finite products, kernels, and ...
- 1,349
29
votes
3
answers
3k
views
The non-simplicity of $SO(4)$ and $A_4$
It is well known that the alternating group $A_n$ is simple unless $n=4$. It is likewise well known that the special orthogonal group $SO(n)$ is essentially simple unless $n=4$ (specifically, the ...
- 3,761
14
votes
0
answers
913
views
The symmetric group and the field with one element
I've heard a few times that the symmetric group is an algebraic group over a field with one element, and that the alternating group is quite specifically $SO_n(\mathbb{F}_1)$. This does make a lot of ...
- 20.2k
16
votes
1
answer
784
views
Are there F_un Lie algebras ?
Background See WP-article on F_1 = F_{un} = Field with one element (and also this MO question). Paraphrasing someone:
we do not know what is it, but it is not a field :). For this question it is ...
- 24.8k
3
votes
1
answer
323
views
affine and projective schemes over $\mathbf{F}_1$?
What should affine and projective schemes over $\mathbf{F}_1$ be?
user19475
54
votes
2
answers
4k
views
Connections between various generalized algebraic geometries (Toen-Vaquié, Durov, Diers, Lurie)?
As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general:
$\bullet$ In the approach by ...
- 63.1k