### Navigation

- index
- modules |
- next |
- previous |
- Combinatorics »
- Crystals »

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

See also

- `sage.categories.crystals`
- `sage.combinat.crystals.crystals`

## Catalog

This is a catalog of crystals that are currently in Sage:

- `AffineCrystalFromClassical`
- `AffineCrystalFromClassicalAndPromotion`
- `AffineFactorization`
- `AlcovePaths`
- `FastRankTwo`
- `GeneralizedYoungWalls`
- `HighestWeight`
- `KirillovReshetikhin`
- `KyotoPathModel`
- `Letters`
- `LSPaths`
- `NakajimaMonomials`
- `ProjectedLevelZeroLSPaths`
- `RiggedConfigurations`
- `Spins`
- `SpinsPlus`
- `SpinsMinus`
- `Tableaux`

Functorial constructions:

- `DirectSum`
- `TensorProduct`

Subcatalogs:

- \(B(\infty)\) `(infinity) crystals`
- `Elementary crystals`
- `Kirillov-Reshetihkin crystals`

### Navigation

- index
- modules |
- next |
- previous |
- Combinatorics »
- Crystals »