Catalog Of Crystals

Let \(I\) be an index set and let \((A,\Pi,\Pi^\vee,P,P^\vee)\) be a Cartan datum associated with generalized Cartan matrix \(A = (a_{ij})_{i,j\in I}\). An abstract crystal associated to this Cartan datum is a set \(B\) together with maps

\[e_i,f_i \colon B \to B \cup \{0\}, \qquad \varepsilon_i,\varphi_i\colon B \to \ZZ \cup \{-\infty\}, \qquad \mathrm{wt}\colon B \to P,\]

subject to the following conditions:

  1. \(\varphi_i(b) = \varepsilon_i(b) + \langle h_i, \mathrm{wt}(b) \rangle\) for all \(b \in B\) and \(i \in I\);
  2. \(\mathrm{wt}(e_ib) = \mathrm{wt}(b) + \alpha_i\) if \(e_ib \in B\);
  3. \(\mathrm{wt}(f_ib) = \mathrm{wt}(b) - \alpha_i\) if \(f_ib \in B\);
  4. \(\varepsilon_i(e_ib) = \varepsilon_i(b) - 1\), \(\varphi_i(e_ib) = \varphi_i(b) + 1\) if \(e_ib \in B\);
  5. \(\varepsilon_i(f_ib) = \varepsilon_i(b) + 1\), \(\varphi_i(f_ib) = \varphi_i(b) - 1\) if \(f_ib \in B\);
  6. \(f_ib = b'\) if and only if \(b = e_ib'\) for \(b,b' \in B\) and \(i\in I\);
  7. if \(\varphi_i(b) = -\infty\) for \(b\in B\), then \(e_ib = f_ib = 0\).