Index of Notation
#m,p, Schubert number, 5
→, rational map, 79
, cover in a poset, 91, 126
A, support of a polynomial, 3, 26
A+, lift of A, 30
α(κ), sequence associated to a subset
κ [m], 174
β(κ), sequence associated to a subset
κ [m], 175
(
[m+p]
p
)
, Bruhat order, 97
, Schubert condition (m, m+2,. . . , m+p),
117, 122
C, complex numbers, 1
C∗,
nonzero complex numbers, 1
Cm,p, product of chains, 96
C{t}, field of Puiseaux series, 34
C[X], homogeneous coordinate ring of
projective variety X, 30
d(A), number of solutions to system with
support A, 26
d(α), degree of Schubert variety Xα, 128
deg(X), degree of subvariety X, 29
ΔA, convex hull of A, 3, 26
Δω, regular polyhedral subdivision induced
by ω, 42
Δp, positive chamber of hyperplane
complement, 65
des(w), descent of Grassmannian condition
w, 183
d(f1, . . . , fn), number of solutions to
f1 = · · · = fn = 0, 26
eα, basis element of
∧pCm+p,
96
E•(s), flag of polynomials vanishing to
different orders at s, 131
F•, flag of subspaces, 117
F•(t), osculating flag, 119, 121
F (a; m), flag manifold, 180
G, linear algebraic group, 179
g+,
lift of map g to sphere, 80
G/P , flag manifold, 179
γ, rational normal curve, 6, 118, 121
Gr(m, Cm+p−1[t]), Grassmannian, 5
Gr(p, m+p), Grassmannian, 96, 116
H, hyperplane arrangement, 63
HX (d), Hilbert function of projective
variety X, 31
hX
(d), Hilbert polynomial, 30
Im,p, Pl¨ ucker ideal, 98
inω(F ), facial system, 33
inω(f), initial form, 33, 40
inω(XA), initial scheme, 40
Ka, Kostka number, 143
κ = κ1 + · · · + κk, sum of a sequence, 174
λ(P ), number of linear extensions of poset
P , 92
LG(m), Lagrangian Grassmannian, 175
lg(w), length of a permutation, 103, 179
mdeg(ρ), mapping degree of ρ, 19
MH, complement of hyperplane
arrangement, 63
m(ω, A), minimum value of ω on A, 33
MV(K1, . . . , Kn), mixed volume of convex
bodies K1, . . . , Kn, 3, 27
N, natural numbers, 1
[n], set of integers {1, 2, . . . , n}, 2
OG(m), orthogonal Grassmannian, 173
OP , order polytope of poset P , 92
PA,
projective space with coordinates
indexed by A, 27
ϕA, monomial parameterization, 27
Pn,
complex projective space, 2
Pω, lifted polytope, 42
ψ, affine-linear map, 63
Q, rational numbers, 1
R, real numbers, 1
R∗,
nonzero real numbers, 1
R , positive real numbers, 1, 65
Rn
, positive orthant, 49
ρ, rational function, 19
195
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