Questions tagged [crystals]
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53 questions
7
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1
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Classification of modules all whose weight spaces are $1$-dimensional
In type $A$, the simple modules all of whose weight spaces are $1$-dimensional are the $L(n\varpi_1)$ and $L(\varpi_k)$. This can be seen from the fact that dimensions of weight spaces are given by ...
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4
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0
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117
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Is there an "$\ell$-adic Riemann Hilbert correspondence"?
The Riemann-Hilbert correspondence (see, e.g., Thm. 7.2.2 of D-modules, perverse sheaves, and representation theory) shows that analytic perverse sheaves are equivalent to regular holonomic $D$-...
- 615
4
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0
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182
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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 ...
- 371
1
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0
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56
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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 ...
- 2,907
3
votes
1
answer
212
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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 ...
- 2,907
1
vote
0
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52
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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 ...
- 2,907
4
votes
1
answer
334
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 ...
- 2,907
2
votes
0
answers
109
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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 ...
- 2,907
1
vote
0
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88
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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 ...
- 2,907
2
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0
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63
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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 $F$-...
- 2,907
4
votes
0
answers
270
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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 ...
- 426
4
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0
answers
111
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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
$$\lambda_1\geq \mu_1\geq \dots \geq \mu_{n-1}\geq \...
2
votes
1
answer
314
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Open/closed embeddings and the de Rham space
Let $U\to X$ be an open immersion of schemes and denote by $D$ the (say reduced) complement. Then by applying the de Rham functor, we get morphisms
$$U_{dR}\to X_{dR}\leftarrow D_{dR}$$
of the ...
- 4,386
5
votes
0
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128
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Classification of connected finite affine type A crystals
In the survey https://www.aimath.org/WWN/kostka/crysdumb.pdf the following statement is stated as a Conjecture 4.5 (due to Kashiwara): "Every connected affine crystal graph is isomorphic to a ...
- 163
5
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0
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110
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Pushforward of crystals in mixed/positive characteristic
Is there a good reference for pushforward of crystals along smooth maps in mixed/positive characteristic with respect to the crystalline site? Intuitively I'm confused on what the pushforward looks ...
- 2,346
4
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0
answers
121
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Looking for a crystalline analogue of , $\mathcal {Z}_{\sim}^* (X)_F \simeq \mathcal{Z}_{\sim}^* (X_{k ' })_F^{\mathrm{Gal} ( k ' / k )} $
Is there a crystalline analogue for the following formula, using the crystalline Frobenius $ F_v $ instead of the absolute Galois group $ \mathrm{Gal} (\overline{k} / k) $ ?
Here is the formula, which ...
- 595
2
votes
0
answers
307
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Looking for the exact and the precise statement of Ogus conjecture
I have been looking for several weeks for the exact and the precise statement of Ogus conjecture, but, I cannot find it.
The only book which made me discover the statement of this conjecture is that ...
- 595
1
vote
0
answers
302
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Berthelot-Ogus comparison isomorphism
On the link, page, $ 2 $, the Berthlot-Ogus isomorphism theorem is stated as follows,
We have a canonical isomorphism, $$ \rho_{\mathrm{cris}} \ : \ H_{\mathrm{cris}}^{i} (X) \otimes_{K_ {0}} K \to H_{...
- 595
3
votes
0
answers
79
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 ...
- 550
13
votes
0
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195
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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 ...
- 820
17
votes
1
answer
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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 ...
- 11.8k
10
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0
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233
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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 $\operatorname{GL}(0)$, ...
- 27.8k
3
votes
0
answers
262
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 ...
- 31
4
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0
answers
150
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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 ...
- 51
6
votes
2
answers
298
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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 ...
- 15.8k
5
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0
answers
535
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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. ...
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3
votes
1
answer
494
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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 ...
- 123
7
votes
0
answers
572
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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 ...
- 371
5
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0
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96
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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 ...
- 2,581
7
votes
2
answers
957
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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 ...
- 607
5
votes
0
answers
460
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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 ...
- 320
4
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0
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195
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(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 ...
- 15.8k
2
votes
1
answer
148
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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 ...
- 2,223
2
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1
answer
403
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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, ...
- 6,201
5
votes
3
answers
791
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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 ...
- 867
10
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1
answer
435
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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, ...
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19
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2
answers
3k
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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 ...
- 5,670
9
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0
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257
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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 ...
- 12.1k
3
votes
1
answer
223
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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 ...
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1
vote
1
answer
441
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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 ...
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2
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0
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203
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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 ...
- 271
2
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0
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473
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$\sigma$-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}...
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28
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4
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3k
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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 ...
user6976
12
votes
1
answer
834
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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 ...
- 8,559
2
votes
1
answer
405
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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 $p>0$ (...
- 637
22
votes
1
answer
1k
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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 ...
- 2,399
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0
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419
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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}...
- 44.7k
18
votes
3
answers
3k
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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 $W_k$ ...
- 4,015
21
votes
1
answer
4k
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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 ...
- 1,500
19
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1
answer
2k
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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 ...
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