Notation The notation used in the book is organized by topic. The number in parentheses at the end of an entry indicates the chapter in which the notation first appears. Basic Sets Z, Q, R, C integers, rational numbers, real numbers, complex numbers N semigroup of nonnegative integers {0,1,2,...} The Torus C∗ multiplicative group of nonzero complex numbers C \{0} (1) (C∗)n standard n-dimensional torus (1) M, χm character lattice of a torus and character of m ∈ M (1) N, λu lattice of one-parameter subgroups of a torus and one-parameter subgroup of u ∈ N (1) TN torus N ⊗Z C∗ = HomZ(M,C∗) associated to N and M (1) MR, MQ vector spaces M ⊗Z R, M ⊗Z Q built from M (1) NR, NQ vector spaces N ⊗Z R, N ⊗Z Q built from N (1) m,u pairing of m ∈ M or MR with u ∈ N or NR (1) Hyperplanes and Half-Spaces Hm hyperplane in NR defined by m,− = 0, m ∈ MR \{0} (1) H+ m half-space in NR defined by m,− ≥ 0, m ∈ MR \{0} (1) Hu,b hyperplane in MR defined by −,u = b, u ∈ NR \{0} (2) H+ u,b half-space in MR defined by −,u≥ b, u ∈ NR \{0} (2) xv

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