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Crystals and rigged configurations

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Author(s): Travis Scrimshaw
Series: PhD thesis at University of California, Davis
Year: 2015

Language: English

1 Introduction 1
1.1 Physics and mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Summary of the main results . . . . . . . . . . . . . . . . . . . . . . . 10
2 Background 13
2.1 Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Simple subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Rigged configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Virtual crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 U 1
q pgq-rigged configurations . . . . . . . . . . . . . . . . . . . . . . . .
32
2.6 Kirillov-Reshetikhin crystals . . . . . . . . . . . . . . . . . . . . . . . 36
2.7 The (virtual) Kleber algorithm . . . . . . . . . . . . . . . . . . . . . 42
3 Generalizing rigged configurations 44
3.1 Rigged configuration model for Bp8q in simply-laced finite type . . . 44
3.2 Extending Theorem 3.1.9 to arbitrary simply-laced Kac–Moody algebras 53
3.3 Extending Theorem 3.1.9 to non-simply-laced Lie algebras . . . . . . 56
3.3.1 Extending Theorem 3.1.9 to all finite types . . . . . . . . . . . 56
3.3.2 Recognition Theorem . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Projecting from RCp8q to RCpλq . . . . . . . . . . . . . . . . . . . . 68
4 Rigged configurations and KR tableaux 72
4.1 Crystal operators on rigged configurations in non-simply-laced types . 72
4.1.1 Virtualization map . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.2 Crystal operators . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.2 The filling map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.1 Filling map for type D p1q
n
. . . . . . . . . . . . . . . . . . . . . 78
4.2.2 Filling map for type C p1q
n
. . . . . . . . . . . . . . . . . . . . . 83
4.2.3 Filling map for type A p2q
2n?1
. . . . . . . . . . . . . . . . . . . . 89
4.2.4 Filling map for type B p1q
n
. . . . . . . . . . . . . . . . . . . . . 91
4.2.5 Filling map for type A p2q
2n
. . . . . . . . . . . . . . . . . . . . . 95
4.2.6 Filling map for type A p2q:
2n
. . . . . . . . . . . . . . . . . . . . 96
4.2.7 Filling map for type D p2q
n?1
. . . . . . . . . . . . . . . . . . . . 97
4.2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3 Affine crystal strucutre . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3.1 Affine crystal operators . . . . . . . . . . . . . . . . . . . . . . 102
4.3.2 Virtualization as affine crystals . . . . . . . . . . . . . . . . . 109
4.3.3 Extension to r ? n . . . . . . . . . . . . . . . . . . . . . . . . 111
4.4 The virtualization map and Φ . . . . . . . . . . . . . . . . . . . . . . 112
4.4.1 Single tensor factors . . . . . . . . . . . . . . . . . . . . . . . 112
4.4.2 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5 Future work 118
5.1 Extensions and future work . . . . . . . . . . . . . . . . . . . . . . . 118
A Extension of Theorem 2.4.8 123
B Calculations using Sage 136