Acknowledgments ................................................ ix
1 Introduction ................................................. 1
2 Introduction to on-shell functions and diagrams .............. 5
2.1 On-shell particles, functions, and kinematical data ..... 5
2.2 Scattering amplitudes and their amalgamations ........... 8
2.3 On-shell building blocks: the massless three-particle
S-matrix ............................................... 10
2.4 On-shell supersymmetry and (maximally) supersymmetric
theories ............................................... 14
2.5 Building up diagrams with "BCFW-bridges" ............... 18
2.6 On-shell recursion for all-loop amplitudes ............. 20
2.7 Physical equivalences among on-shell diagrams .......... 23
3 Permutations and scattering amplitudes ...................... 29
3.1 Combinatorial descriptions of scattering processes ..... 29
3.2 The BCFW-bridge construction of representative graphs .. 33
4 From on-shell diagrams to the Grassmannian .................. 37
4.1 The Grassmannian of k-planes in n dimensions, G(k, n) .. 37
4.2 Grassmannian description of kinematical data - λ and
λ ...................................................... 40
4.3 Grassmannian representation of on-shell diagrams ....... 41
4.4 Amalgamation of on-shell diagrams ...................... 45
4.5 "Boundary measurements" and canonical coordinates ...... 49
4.6 Coordinate transformations induced by moves and
reduction .............................................. 52
4.7 Relation to composite leading singularities ............ 56
5 Configurations of vectors and the positive Grassmannian ..... 60
5.1 The geometry and combinatorics of the positroid
stratification ......................................... 60
5.2 Canonical coordinates and the equivalence of
permutation labeling ................................... 66
5.3 Positroid cells and the positive part of the
Grassmannian ........................................... 69
5.4 Canonically positive coordinates for positroids ........ 73
6 Boundary configurations, graphs, and permutations ........... 77
6.1 Physical singularities and positroid boundaries ........ 77
6.2 Boundary configurations: combinatorics and
stratification ......................................... 78
6.3 (Combinatorial) polytopes in the Grassmannian .......... 79
6.4 Approaching boundaries in canonical coordinates ........ 81
7 The invariant top-form and the positroid stratification ..... 83
7.1 Equivalence with the canonical positroid volume form ... 85
8 (Super-)conformal and dual conformal invariance ............. 88
8.1 The Grassmannian geometry of momentum conservation ..... 88
8.2 Twistor space and the superconformality of on-shell
forms .................................................. 90
8.3 Momentum-twistors and dual super-conformal invariance .. 92
9 Positive diffeomorphisms and Yangian invariance .......... 97
10 The kinematical support of physical on-shell forms ......... 101
10.1 Kinematical support of NMHV Yangian-invariants ........ 102
10.2 Kinematical support for one-dimensional kinematics .... 103
10.3 A general combinatorial test of kinematical support ... 103
11 Homological identities among Yangian-invariants ............ 107
11.1 Homological identities in the Grassmannian ............ 108
12 (Relatively) orienting canonical coordinate charts on
positroids configurations .................................. 113
12.1 Comparing the orientations of canonical coordinate
charts ................................................ 115
12.2 Accessing boundary configurations from the canonical
atlas ................................................. 118
13 Classification of Yangian-invariants and their relations ... 121
14 The Yang-Baxter relation and ABJM theories ................. 127
14.1 The on-shell avatar of the Yang-Baxter relation ....... 127
14.2 ABJM theories ......................................... 130
15 On-shell diagrams for theories with  < 4
supersymmetries ............................................ 136
16 Dual graphs and cluster algebras ........................... 141
16.1 The 'dual' of an on-shell diagram ..................... 141
16.2 Cluster algebras: seeds, mutations, and cluster
coordinates ........................................... 146
16.3 Cluster amalgamation .................................. 150
16.4 (Brief) overview of cluster structures in physics ..... 152
17 On-shell representations of scattering amplitudes .......... 155
17.1 (Diagrammatic) proof of the BCFW recursion relations .. 156
17.2 The structure of (tree) amplitudes in the
Grassmannian .......................................... 159
17.3 Canonical coordinates for loop integrands ............. 164
17.4 The transcendentality of loop amplitudes .............. 173
18 Outlook .................................................... 178
References ................................................. 183
Index ......................................................... 192
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