Skip to main content

Questions tagged [crystals]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
4 votes
0 answers
165 views

Étale- or fppf-crystalline sites

I have a straightforward question. Let (say) $X/\mathbb{F}_p$ be a smooth proper scheme. On the big crystalline category over $\mathbb{Z}/p^n$ one can take the Zariski or étale topology, and one can ...
Curious's user avatar
  • 341
1 vote
0 answers
45 views

Frobenius pullback of an integrable connection on a quasi-projective scheme

Let $X_k$ be a smooth quasi-projective scheme over a finite field $k$. Let $X_K$ be a smooth lift of $X_k$ to characteristic $0$, and let $(X_K)^{\text{an}}$ denote the rigid analytic space associated ...
kindasorta's user avatar
  • 2,103
3 votes
1 answer
197 views

Isocrystal with no $F$-structure

$\DeclareMathOperator\Isoc{Isoc}$Let $X_k$ be a quasiprojective $k$ scheme, with $k$ finite, and let $X_K$ be the rigid analytic space lifting it to the fraction field of its Witt ring, which I denote ...
kindasorta's user avatar
  • 2,103
1 vote
0 answers
50 views

Frobenius acting by autoequivalence on $\text{Isoc}(X/K)$

Let $X_k$ be a smooth quasiprojective scheme over a finite field $k$. Let $X_K$ be a smooth lift of $X_k$ to the fraction field of the Witt ring of $k$, which I denote by $K$. In various papers I read ...
kindasorta's user avatar
  • 2,103
4 votes
1 answer
247 views

Equivalence between vector bundles with integrable connections to isocrystals

Let $k$ be a perfect field, $W(k)$ its Witt ring, and $K$ the fraction field of $W(k)$. Let $X_k$ be a smooth proper curve over $k$, and let $X_K$ be the schematic generic fibre of a smooth proper ...
kindasorta's user avatar
  • 2,103
2 votes
0 answers
99 views

Extensions of $F$-isocrystals

Let $X$ be a smooth affine scheme over $k$, a finite field. Let $W(k)$ denote the Witt ring, and $K$ its fraction field. Fix a smooth lift of $X$ to $K$ and denote it by $X_K$. Let $b\in X(k)$ denote ...
kindasorta's user avatar
  • 2,103
1 vote
0 answers
82 views

Extensions in the category $F\text{-Isoc}(X)$

Let $X$ be a smooth affine scheme over a finite field $k$, let $W(k)$ denote its Witt ring, and by $K$ its fraction field. Let $F\text{-Isoc}(X/K)$ denote the category of convergent $F$-isocrystals on ...
kindasorta's user avatar
  • 2,103
2 votes
0 answers
61 views

Fibre functors of the category $F\text{-Isoc}(X)$

Let $X$ be a smooth affine scheme over a finite field $k$. Denote its Witt ring by $W(k)$, and the fraction field of its Witt ring by $K$. Let $F\text{-Isoc}(X)$ denote the category of convergent $F$-...
kindasorta's user avatar
  • 2,103
3 votes
0 answers
242 views

Confusion about definition of crystals

In the notes by Lurie there seems to be two possible definition for crystals which both makes sense for arbitrary functors $X : \mathrm{CRing}_k \to \mathrm{Set}$. ($k$ here is a field. We probably ...
user577413's user avatar
4 votes
0 answers
103 views

How many diagrams interlace a given Young diagram?

For a fixed partition $\lambda=(\lambda_1\geq\dots\geq \lambda_n)$ we say $\mu=(\mu_1\geq \dots \geq \mu_{n-1})$ $\textit{interlaces}$ $\lambda$ iff $$\lambda_1\geq \mu_1\geq \dots \geq \mu_{n-1}\geq \...
Nicolas Medina Sanchez's user avatar
2 votes
1 answer
297 views

Open/closed embeddings and the de Rham space

Let $U\to X$ be an open immersion of schemes and denote by $D$ the (say reduced) complement. Then by applying the de Rham functor, we get morphisms $$U_{dR}\to X_{dR}\leftarrow D_{dR}$$ of the ...
curious math guy's user avatar
5 votes
0 answers
123 views

Classification of connected finite affine type A crystals

In the survey https://www.aimath.org/WWN/kostka/crysdumb.pdf the following statement is stated as a Conjecture 4.5 (due to Kashiwara): "Every connected affine crystal graph is isomorphic to a ...
Asav's user avatar
  • 163
5 votes
0 answers
107 views

Pushforward of crystals in mixed/positive characteristic

Is there a good reference for pushforward of crystals along smooth maps in mixed/positive characteristic with respect to the crystalline site? Intuitively I'm confused on what the pushforward looks ...
davik's user avatar
  • 2,035
4 votes
0 answers
121 views

Looking for a crystalline analogue of , $\mathcal {Z}_{\sim}^* (X)_F \simeq \mathcal{Z}_{\sim}^* (X_{k ' })_F^{\mathrm{Gal} ( k ' / k )} $

Is there a crystalline analogue for the following formula, using the crystalline Frobenius $ F_v $ instead of the absolute Galois group $ \mathrm{Gal} (\overline{k} / k) $ ? Here is the formula, which ...
Angel65's user avatar
  • 595
2 votes
0 answers
293 views

Looking for the exact and the precise statement of Ogus conjecture

I have been looking for several weeks for the exact and the precise statement of Ogus conjecture, but, I cannot find it. The only book which made me discover the statement of this conjecture is that ...
Angel65's user avatar
  • 595
1 vote
0 answers
276 views

Berthelot-Ogus comparison isomorphism

On the link, page, $ 2 $, the Berthlot-Ogus isomorphism theorem is stated as follows, We have a canonical isomorphism, $$ \rho_{\mathrm{cris}} \ : \ H_{\mathrm{cris}}^{i} (X) \otimes_{K_ {0}} K \to H_{...
Angel65's user avatar
  • 595
3 votes
0 answers
75 views

Multiplicity relation between highest weight modules, Demazure modules, and crystals

Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra, and let $\lambda$ be an associated dominant integral weight. Then two different objects we can relate to this data is $V(\lambda)$, the ...
SamJeralds's user avatar
13 votes
0 answers
191 views

Relationship between crystal root operators and usual $e_i, f_i$?

Suppose I am working in a symmetrizable Kac–Moody Lie algebra $\mathfrak{g}$. Let $e_1,\dotsc,e_n,f_1,\dotsc,f_n$ denote the usual Chevalley generators of $\mathfrak{g}$. Let $V$ be a highest weight ...
ArB's user avatar
  • 688
16 votes
1 answer
930 views

On a series of lectures of Deligne on crystalline cohomology in characteristic $0$

In the introduction of Berthelot's book on crystalline cohomology [Ber74], one finds, on page 11, the following passage: i) des travaux de P. Deligne ([14], non publiés) prouvant en particulier le ...
Emily's user avatar
  • 11.5k
10 votes
0 answers
209 views

Branching from GL(a+b) to GL(a) x GL(b)$ using Gel'fand-Cetlin patterns

If one iterates the multiplicity-free branching rule from $\operatorname{GL}(n)$ representations (finite-dim, over $\mathbb C$) to $\operatorname{GL}(n-1)$ all the way down to $\operatorname{GL}(0)$, ...
Allen Knutson's user avatar
3 votes
0 answers
256 views

Kashiwara's definition of normal crystal

Let $\mathfrak{g}$ be a symmetrisable Kac-Moody algebra, and $U_q(\mathfrak{g})$ its associated quantum group. Each integrable module of $U_q(\mathfrak{g})$ admits a crystal basis, as was first shown ...
Joel's user avatar
  • 31
4 votes
0 answers
147 views

Crystals and nilpotence

Fix a prime $p$ and a height $n \geq 1$, then there is a closed substack $\mathcal{M}^{\geq n}$ of the stack of $\mathbb{F}_p$-formal groups consisting of formal groups having height $\geq n$. A ...
crystal's user avatar
  • 51
6 votes
2 answers
287 views

RSK and crystal operators

Is there a good reference on how RSK (and the 3 other variants) interact with crystal operators on the semi-standard tableaux $(P,Q)$ in the image? That is, we have biwords, $W$ which are in ...
Per Alexandersson's user avatar
5 votes
0 answers
483 views

Basic questions about crystals and Grothendieck connections

I have a few basic questions about Grothendieck connections and crystals. I know it's bad practice to ask a bunch of different questions at once, however I feel they naturally come bundled together. ...
Arrow's user avatar
  • 10.4k
3 votes
1 answer
482 views

Calculating cohomology group $H^3(point group,\mathbb{Z})$ using GAP program

I'm trying to compute $H^3(point group,\mathbb{Z})$ for all the 32 point groups in 3D which has some applications in physics. Unfortunately, I could not find literature discussing this problem. So I ...
Xu Yang's user avatar
  • 123
7 votes
0 answers
550 views

Dieudonne modules vs Dieudonne crystals reference/clarification

I've read a bit about Dieudonné modules, mainly from Fontaine's "Groupes p-divisibles sur les corps locaux" and Demazure's "Lectures on $p$-divisible groups". I am familiar with the main ...
aytio's user avatar
  • 371
5 votes
0 answers
94 views

Is every space group the symmetry group of some triply periodic minimal surface?

I know that there are a lot of TPMS with different symmetry groups. It seems like every space group is the symmetry group of some TPMS. But I can not find a reference that confirms this for all the ...
Hao Chen's user avatar
  • 2,541
7 votes
2 answers
933 views

Is this a quasi-crystal and/or a fractal?

I'm not too familiar with quasi-crystals, but I was recently playing around with a particular discrete function and I got the following neat pattern: This is just a small segment, but as far as I ...
Ruben Verresen's user avatar
5 votes
0 answers
451 views

Relation between crystalline and perverse sheaves

Take $X$ to be a smooth complex projective algebraic variety. The Riemann-Hilbert correspondence gives an equivalence of categories between the category of perverse sheaves on $X$ and the category of ...
Ben Morley's user avatar
4 votes
0 answers
194 views

(Double) Crystal reflection operators on SSYTs

I am not that familiar with the language of crystals, but this is what I know: Let $SSYT(\lambda, \mu)$ be the set of semi-standard Young tableaux with shape $\lambda$ and weight $\mu$. There are ...
Per Alexandersson's user avatar
2 votes
1 answer
146 views

positions of regular cubes in Euclidean space with all its vertices without distinction

Let $P$ be the standard regular cube, centered at the origin of $\mathbb{R}^3$ with all its vertices on the unit sphere. If the vertices are labelled by $1,2,3,4,5,6,7,8$, then the collection of all ...
QSR's user avatar
  • 2,223
2 votes
1 answer
394 views

Two questions about Whittaker functions

I am watching the video: Modeling p-adic Whittaker functions, Part I. I have two questions about Whittaker functions in the video. From 33:00 to 37:00, it is said that after changing of variables, ...
Jianrong Li's user avatar
  • 6,141
5 votes
3 answers
770 views

Crystal structure, lattice, Graph and coloring

I am working across mathematics, physics and engineering. And I am looking for whether there exists already formally established knowledge in the field. Given a periodic graph (actually a physical ...
user40780's user avatar
  • 867
10 votes
1 answer
434 views

Does the 'string property' finish Joseph's proof of Demazure character formula?

The too long, didn't read form of the question would simply be, has someone completed A. Joseph's proof of the Demazure character formula? Is Joseph's proof considered complete? In more detail, ...
denomme's user avatar
  • 101
19 votes
2 answers
3k views

Letter from Grothendieck to Tate on "crystals"

I have downloaded from this link a quite poor quality scan of the letter dating May 1966 that Grothendieck sent to Tate mentioning his ideas about generalizing Monsky-Washnitzer cohomology. I am ...
Filippo Alberto Edoardo's user avatar
9 votes
0 answers
251 views

Good bounds for the number of $n$-dimensional crystallographic groups ?

Let $s(n)$ denote the number of distinct crystallographic groups in $Isom(\mathbb{R}^n)$. Apparently the best known upper bound so far is $$ s(n)\le e^{e^{4n^2}}, $$ given by Peter Buser in $1985$. On ...
Dietrich Burde's user avatar
3 votes
1 answer
217 views

Can elements of Weil algebras be detected by maps into truncated symmetric algebras?

Recall that a Weil algebra is a finite-dimensional real unital algebra that admits exactly one homomorphism to R. Such algebras form the basis of the Weil approach to differential geometry, pioneered ...
Dmitri Pavlov's user avatar
1 vote
1 answer
433 views

Galois descent for semilinear endomorphisms

Let $K \subset L$ be a finite Galois extension, $\sigma$ an automorphism of $L$ (not necessarily fixing $K$) and let $E$ be a finite-dimensional vector space over $L$ together with an $\sigma$-linear ...
user26756's user avatar
  • 271
2 votes
0 answers
202 views

Two different definitions of $\sigma$-L-spaces in Kottwitz I and II

In his papers "Isocrystals with additional structure" I and II, Kottwitz defines the notion of $\sigma$-$L$-spaces. In the first one the situation is the following $k$ an algebraically closed field ...
user26756's user avatar
  • 271
2 votes
0 answers
471 views

$\sigma$-conjugate iff $p$-adically close

First some notations. Let $p$ be a prime, $k$ a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vectors over $k$, $\sigma : W \rightarrow W$ the Frobenius, $R$ a commutative $\mathbb{Z}...
user26756's user avatar
  • 271
28 votes
4 answers
3k views

Shape of snowflakes

Is there a mathematical theory that explains the shape of a snowflake? Why is it not round? Update Tree-like metric spaces appear often as limits of sequences of metric spaces (say, asymptotic cones ...
user avatar
12 votes
1 answer
822 views

R-matrices, crystal bases, and the limit as q -> 1

I am seeking references for precise statements and rigorous proofs of some facts about the actions of quantum root vectors and $R$-matrices on crystal bases for finite-dimensional representations of ...
MTS's user avatar
  • 8,479
2 votes
1 answer
402 views

Isomorphism between pull-backs of an F-crystal by different liftings of Frobenius

This might be a naive question. But since I haven't seen this in any reference, I'll try to ask it here. Let $T$ be a smooth scheme over the algebraically closed field $k$ of characteristic $p>0$ (...
Jack's user avatar
  • 637
22 votes
1 answer
1k views

What is classified by the (big) crystalline topos?

In his paper "Generic Galois Theory of Local Rings", G.C. Wraith states on p. 743 that the (big) crystalline topos "can be conveniently described in terms of the theory it describes". What exactly is ...
Marc Nieper-Wißkirchen's user avatar
11 votes
0 answers
412 views

Is there a notion of tensor product of perfect bases of representations of Lie algebras?

Berenstein and Kazhdan define perfect bases as an "unquantized" version of crystal bases. A perfect basis is roughly a basis with a crystal structure such that $E_i\cdot v=\mathbb{C}\cdot \tilde{e}...
Ben Webster's user avatar
  • 44.3k
18 votes
3 answers
3k views

Lifting varieties to characteristic zero.

If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W_k$ ...
Xandi Tuni's user avatar
  • 3,985
21 votes
1 answer
3k views

Crystalline cohomology of abelian varieties

I am trying to learn a little bit about crystalline cohomology (I am interested in applications to ordinariness). Whenever I try to read anything about it, I quickly encounter divided power ...
Martin Orr's user avatar
  • 1,500
18 votes
1 answer
2k views

The Infinitesimal topos in positive characteristic

This question was inspired by and is somewhat related to this question. In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie ...
Lars's user avatar
  • 4,420
6 votes
1 answer
293 views

Are there elements of fixed weight in a crystal not killed by too many Kashiwara operators?

I've come across an annoying lemma trying to finish up an argument, and I was hoping one of you guys knew about it. Question: Given a weight $\lambda$ of a simple Lie algebra $\mathfrak g$, and ...
Ben Webster's user avatar
  • 44.3k
16 votes
2 answers
1k views

Is the tangent space functor from PD formal groups to Lie algebras an equivalence?

The previous version of this question was rather badly broken, and I hope this version makes some sense. There have been a few questions on MathOverflow about how much representation-theoretic ...
S. Carnahan's user avatar
  • 45.3k