Catalog Of Crystals¶

Definition of a Crystal¶

Let $C$ be a CartanType with index set $I$ , and $P$ be the corresponding weight lattice of the type $C$ . Let $\alpha_i$ and $\alpha^{\vee}_i$ denote the corresponding simple roots and coroots respectively. Let us give the axiomatic definition of a crystal.

A type $C$ crystal $\mathcal{B}$ is a non-empty set with maps $\operatorname{wt} : \mathcal{B} \to P$ , $e_i, f_i : \mathcal{B} \to \mathcal{B} \cup \{0\}$ , and $\varepsilon_i, \varphi_i : \mathcal{B} \to \ZZ \cup \{-\infty\}$ for $i \in I$ satisfying the following properties for all $i \in I$ :

$\varphi_i(b) = \varepsilon_i(b) + \langle \alpha^{\vee}_i, \operatorname{wt}(b) \rangle$ ,
if , then:
- $\operatorname{wt}(e_i x) = \operatorname{wt}(b) + \alpha_i$ ,
- $\varepsilon_i(e_i b) = \varepsilon_i(b) - 1$ ,
- $\varphi_i(e_i b) = \varphi_i(b) + 1$ ,
if , then:
- $\operatorname{wt}(f_i b) = \operatorname{wt}(b) - \alpha_i$ ,
- $\varepsilon_i(f_i b) = \varepsilon_i(b) + 1$ ,
- $\varphi_i(f_i b) = \varphi_i(b) - 1$ ,
$f_i b^{\prime} = b$ if and only if $e_i b = b^{\prime}$ for $b, b^{\prime} \in \mathcal{B}$ ,
if $\varphi_i(b) = -\infty$ for $b \in \mathcal{B}$ , then $e_i b = f_i b = 0$ .

See also

Catalog¶

This is a catalog of crystals that are currently in Sage:

Functorial constructions:

Subcatalogs:

Table Of Contents

Catalog Of Crystals
- Definition of a Crystal
- Catalog

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