# 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 $$e_i b \in \mathcal{B}$$, 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 $$f_i b \in \mathcal{B}$$, 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 $$e_i b = f_i b = 0$$.

## Catalog¶

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

Functorial constructions:

Subcatalogs: