 Cocharacter

A $k$-defined cocharacter of an algebraic torus $T$ is a homomorphism of algebraic groupsover $k$. A geometric cocharacter is defined after passing to a separable closure; the cocharacter lattice of an algebraic torus uses these geometric maps with their Galois action. The $k$-defined maps are the Galois-fixed elements.

For $T={\mathbb{G}}_m^2$, an integer pair $(a,b)$ gives $\lambda (t)=(t^a,t^b)$. Composing with the character $\chi (x,y)=x^m{y}^n$ yields $\chi (\lambda (t))=t^{ma+nb}$, defining the integer character-cocharacter pairing. This makes the two geometric lattices dual.

Cocharacters describe algebraic one-parameter subgroups and the integral directions underlying toric cones. For instance, $t\mapsto (t^2,t^3)$ has primitive exponent vector $(2,3)$; Bézout recovers $t$ from the coordinates by $t^3/t^2$. The distinction between geometric and ground-field-defined cocharacters matters for nonsplit tori.