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19
votes
0answers
621 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 ...
11
votes
0answers
324 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 ...
9
votes
0answers
196 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 ...
5
votes
0answers
102 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 ...
2
votes
0answers
367 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 ...
1
vote
0answers
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 ...