Regular Crystals¶
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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