The crystals tag has no usage guidance.

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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 ...

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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 ...

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**1**answer

109 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 ...

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votes

**1**answer

305 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, ...

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334 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 ...

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**1**answer

337 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, ...

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197 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 ...

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**1**answer

170 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 ...

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**1**answer

322 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 ...

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148 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 ...

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368 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}...

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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 ...

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484 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 ...

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**1**answer

311 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 $p>0$ (...

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628 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 ...

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328 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}...

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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 $W_k$ ...

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2k 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 ...

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**1**answer

265 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
...

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757 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 ...

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497 views

### Canonical basis for the extended quantum enveloping algebras

I am trying to understand some construction done by Lusztig in his book on quantum groups. Given some Cartan datum, let $U=U_q(\mathfrak{g})$ the standard quantized enveloping algebra of the Kac-Moody ...