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 \(e_i b = f_i b = 0\).

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