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§1.1. Introduction to Affine Toric Varieties 11 We begin with characters and one-parameter subgroups. A character of a torus T is a morphism χ : T → C∗ that is a group homo- morphism. For example, m = (a1,...,an) ∈ Zn gives a character χm : (C∗)n → C∗ defined by (1.1.1) χm(t 1 ,...,tn) = t a1 1 ···t an n . One can show that all characters of (C∗)n arise this way (see [152, §16]). Thus the characters of (C∗)n form a group isomorphic to Zn. For an arbitrary torus T, its characters form a free abelian group M of rank equal to the dimension of T. It is customary to say that m ∈ M gives the character χm : T → C∗. We will need the following result concerning tori (see [152, §16] for a proof). Proposition 1.1.1. (a) Let T1 and T2 be tori and let Φ : T1 → T2 be a morphism that is a group homo- morphism. Then the image of Φ is a torus and is closed in T2. (b) Let T be a torus and let H ⊆ T be an irreducible subvariety of T that is a subgroup. Then H is a torus. Assume that a torus T acts linearly on a finite dimensional vector space W over C, where the action of t ∈ T on w ∈ W is denoted t · w. A basic result is that the linear maps w → t · w are diagonalizable and can be simultaneously diagonalized. We describe this as follows. Given m ∈ M, define the eigenspace Wm = {w ∈ W | t · w = χm(t)w for all t ∈ T}. If Wm = {0}, then every w ∈ Wm \{0} is a simultaneous eigenvector for all t ∈ T, with eigenvalue given by χm(t). See [257, Thm. 3.2.3] for a proof of the following. Proposition 1.1.2. In the above situation, we have W = m∈M Wm. A one-parameter subgroup of a torus T is a morphism λ : C∗ → T that is a group homomorphism. For example, u = (b1,...,bn) ∈ Zn gives a one-parameter subgroup λu : C∗ → (C∗)n defined by (1.1.2) λu(t) = (t b1 ,...,t bn ). All one-parameter subgroups of (C∗)n arise this way (see [152, §16]). It follows that the group of one-parameter subgroups of (C∗)n is naturally isomorphic to Zn. For an arbitrary torus T, the one-parameter subgroups form a free abelian group N of rank equal to the dimension of T. As with the character group, an element u ∈ N gives the one-parameter subgroup λu : C∗ → T . There is a natural bilinear pairing , : M × N → Z defined as follows. • (Intrinsic) Given a character χm and a one-parameter subgroup λu, the com- position χm ◦ λu : C∗ → C∗ is a character of C∗, which is given by t → t for some ∈ Z. Then m,u = .