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:
- \(\varphi_i(b) = \varepsilon_i(b) + \langle h_i, \mathrm{wt}(b) \rangle\) for all \(b \in B\) and \(i \in I\);
- \(\mathrm{wt}(e_ib) = \mathrm{wt}(b) + \alpha_i\) if \(e_ib \in B\);
- \(\mathrm{wt}(f_ib) = \mathrm{wt}(b) - \alpha_i\) if \(f_ib \in B\);
- \(\varepsilon_i(e_ib) = \varepsilon_i(b) - 1\), \(\varphi_i(e_ib) = \varphi_i(b) + 1\) if \(e_ib \in B\);
- \(\varepsilon_i(f_ib) = \varepsilon_i(b) + 1\), \(\varphi_i(f_ib) = \varphi_i(b) - 1\) if \(f_ib \in B\);
- \(f_ib = b'\) if and only if \(b = e_ib'\) for \(b,b' \in B\) and \(i\in I\);
- if \(\varphi_i(b) = -\infty\) for \(b\in B\), then \(e_ib = f_ib = 0\).
Catalog¶
This is a catalog of crystals that are currently implemented in Sage:
AffineCrystalFromClassical
AffineCrystalFromClassicalAndPromotion
AffineFactorization
AffinizationOf
AlcovePaths
FastRankTwo
GeneralizedYoungWalls
HighestWeight
Induced
KacModule
KirillovReshetikhin
KleshchevPartitions
KyotoPathModel
Letters
LSPaths
Minimaj
NakajimaMonomials
OddNegativeRoots
ProjectedLevelZeroLSPaths
RiggedConfigurations
ShiftedPrimedTableaux
Spins
SpinsPlus
SpinsMinus
Tableaux
Subcatalogs:
- Catalog Of Crystal Models For
- Catalog Of Elementary Crystals
- Catalog Of Crystal Models For Kirillov-Reshetikhin Crystals
Functorial constructions: