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Graph parameters and invariants of the orthogonal group
Regts, G.
Publication date
2013
Link to publication
Citation for published version (APA):
Regts, G. (2013). Graph parameters and invariants of the orthogonal group.
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Contents
Preface xi
1 Introduction 1
1.1 Background and motivation . . . 2
1.2 Contributions . . . 4
1.3 Outline of this thesis . . . 4
2 Preliminaries 7 2.1 Some notation and conventions . . . 7
2.2 Labeled graphs and fragments . . . 8
2.2.1 Labeled graphs . . . 8
2.2.2 Fragments . . . 9
2.3 Connection matrices . . . 11
2.4 Graph algebras . . . 12
3 Partition functions of edge- and vertex-coloring models 15 3.1 Graph parameters from statistical models . . . 15
3.2 Partition functions of vertex-coloring models . . . 17
3.3 Partition functions of edge-coloring models . . . 18
3.4 Tensor networks . . . 20
3.5 The orthogonal group . . . 21
3.6 Computational complexity . . . 22
4 Invariant theory 25 4.1 Representations and invariants . . . 25
4.2 FFT and SFT for the orthogonal group . . . 27
4.3 Existence and uniqueness of closed orbits . . . 29
4.4 Proof of the Tensor FFT . . . 30 vii
CONTENTS
5 Characterizing partition functions of edge-coloring models 35
5.1 Introduction . . . 35
5.2 Finite rank edge-coloring models . . . 38
5.2.1 Catalan numbers and the rank of Nf−2,l . . . 40
5.3 Framework . . . 42
5.4 Proof of Theorem 5.3 . . . 46
5.5 Proof of Theorem 5.4 . . . 47
5.6 Analogues for directed graphs . . . 50
6 Connection matrices and algebras of invariant tensors 51 6.1 Introduction . . . 51
6.2 The rank of edge-connection matrices . . . 54
6.2.1 Algebra of fragments . . . 54
6.2.2 Contractions . . . 55
6.2.3 Stabilizer subgroups of the orthogonal group . . . 57
6.2.4 The real case . . . 58
6.2.5 The algebraically closed case . . . 58
6.3 The rank of vertex-connection matrices . . . 60
6.3.1 Another algebra of labeled graphs . . . 60
6.3.2 Some operations on labeled graphs and tensors . . . 62
6.3.3 Proof of Theorem 6.1 . . . 64
6.4 Proofs of Theorem 6.11 and Theorem 6.16 . . . 65
7 Edge-reflection positive partition functions of vertex-coloring models 69 7.1 Introduction . . . 69
7.2 Orbits of vertex-coloring models . . . 73
7.2.1 The one-parameter subgroup criterion . . . 73
7.2.2 Application to vertex-coloring models . . . 75
7.3 Proof of Theorem 7.3 . . . 77
8 Compact orbit spaces in Hilbert spaces and limits of edge-coloring models 83 8.1 Introduction . . . 83
8.2 Compact orbit spaces in Hilbert spaces and applications . . . 85
8.2.1 Compact orbit spaces in Hilbert spaces . . . 86
8.2.2 Application of Theorem 8.2 to graph limits . . . 87
8.2.3 Application of Theorem 8.2 to edge-coloring models . . . 88
8.3 Proof of Theorem 8.2 . . . 90
8.4 Proofs of Theorem 8.3 and 8.4. . . 92
8.4.1 Properties of the map π . . . . 93 viii
CONTENTS 8.4.2 Proof of Theorem 8.4 . . . 96 Summary 99 Samenvatting 101 Bibliography 103 Index 109 List of symbols 113 ix
