Regular Crystals

sage.categories.regular_crystals.RegularCrystals

The category of regular crystals.

A crystal is called regular if every vertex \(b\) satisfies

\[\varepsilon_i(b) = \max\{ k \mid e_i^k(b) \neq 0 \} \quad \text{and} \quad \varphi_i(b) = \max\{ k \mid f_i^k(b) \neq 0 \}.\]

Note

Regular crystals are sometimes referred to as normal. When only one of the conditions (on either \(\varphi_i\) or \(\varepsilon_i\)) holds, these crystals are sometimes called seminormal or semiregular.

EXAMPLES:

sage: C = RegularCrystals()
sage: C
Category of regular crystals
sage: C.super_categories()
[Category of crystals]
sage: C.example()
Highest weight crystal of type A_3 of highest weight omega_1