= One-dimensional rational characters of the general linear group
{title2=$X^*(GL_m)=\mathbb Z\cdot\det$}
Every one-dimensional rational $GL_m$ representation is $\det^r$ for one integer $r$. Its restriction to the diagonal torus is a <Laurent monomial>. Invariance under conjugation by permutation matrices forces all its exponents to agree, and density of <diagonalizable> invertible matrices gives the result on the whole group. For $GL_1$, the <character> identity $p(zw)=p(z)p(w)$ directly forces a <Laurent polynomial> to be one monomial with coefficient one.
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