Combinatorics of Crystal Folding

Introduction......Page 8
Monoidal categories and monoidal functors......Page 17
Cartan data......Page 19
Abstract crystals......Page 22
The Kashiwara-Littelmann crystals......Page 25
The polyhedral crystal......Page 27
Admissible automorphisms......Page 31
Folding and the Langlands dual operation......Page 32
Parabolic subdata adapted to folding......Page 33
Applications of folding......Page 36
Balanced parabolic quotients......Page 38
The balanced parabolic quotient......Page 39
Normal forms for JW in types A and D......Page 40
Enumeration of J W......Page 44
Inversion sets and the balanced quotient......Page 48
Exceptional types......Page 55
Category of -foldable crystals......Page 57
The folded crystal......Page 59
Monoidal category structure......Page 65
Highest-weight elements of the folded crystal......Page 68
Miscellaneous results......Page 73
Folded structure of the Kashiwara-Littelmann crystals......Page 76
The -action on the polyhedral crystal......Page 77
Foldability......Page 79
Highest-weight elements of the KL crystals......Page 81
Connectedness......Page 97
Type (D3, C2)......Page 98