244 INDEX
matrix, 37
extended, 37
relation, 38
Face, 166, 181
bounded, 166, 181
unbounded, 166, 181
weight, 166, 181
Factorization parameters, 7
Flag
standard, 2
variety, 2
complete, 2
generalized, 6
Flip, 116
spun, 125
Freezing (of variables), 43
Gauge
group, 180
transformation
global, 150
local, 150
Gauss factorization, 6
Generalized minor, 6
Graph E6-compatible, 106
Grassmannian, 1
Ground ring, 39
Hamiltonian vector field, 8
Horocycle, 115
Ideal
arcs, 115
compatible, 115
triangle, 115
triangulation, 115
nice, 116
perfect, 116
Index
i-bounded, 28
i-exchangeable, 62
Induced subgraph, 54
Intersection number (modified geometric),
120
spun, 127
Sn-invariance, 127
Involutivity, 127
Jacobi
identity, 8
matrix, 199
Killing form, 3
Lattice
Coxeter–Toda, 200
open Volterra, 230
relativistic Toda, 231
Toda, 200
Laurent phenomenon, 44
Lax equation, 199, 228
Leibniz identity, 8
Lie bialgebra, 10
factorizable, 10
Lie-Poisson bracket, 9
Locality, 127
Log-canonical basis
with respect to Poisson structure, 67
with respect to pre-symplectic structure,
111
Loop erasure, 166
Markov
equation, 42
triple, 42
Matrix
Cartan, 49
2-finite, 50
irreducible, 71, 112
Jacobi, 199
mutation, 38
quasi-Cartan, 49
reducible, 71, 112
sign-skew-symmetric, 37
totally, 38
sign-symmetric, 4
skew-symmetric, 37
skew-symmetrizable, 37
D-skew-symmetrizable, 37
Moment, 207
functional, 207
Moser map, 200
Mutation equivalent
matrices, 38
seeds, 39
Noose, 125
Normalization condition, 43
Orthant, 28
Path, 143
reversal, 192
map, 192
weight, 143
Penner coordinates, 116
Perfect planar network
in a disk, 142
in an annulus, 176
Pl¨ucker
coordinates, 1
crossing, 15
non-crossing, 15
stable, 16
embedding, 1
relations, 1
short relations, 2, 15
Poisson
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