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.

Table Of Contents

Previous topic

Crystals

Next topic

Catalog Of Elementary Crystals

This Page