= Solution
Work over $\mathbb C$ with a finite-dimensional <semisimple Lie algebra> $\mathfrak g$. Choose a <Cartan subalgebra> $\mathfrak h$, its <root system> $R\subset\mathfrak h^*$, a positive system $R^+$ and its <simple roots>. Let $W$ be the <Weyl group>, generated by the corresponding root reflections, and put $\rho=\tfrac12\sum_{\alpha\in R^+}\alpha$, the <Weyl vector>. The finite-dimensional irreducible <highest-weight representation> $L(\lambda)$ is indexed by a <dominant integral weight> $\lambda$.
If $P$ is the <weight lattice>, its <group ring> has formal symbols $e^\mu$ with multiplication $e^\mu e^\nu=e^{\mu+\nu}$. The <formal character> is
$$
\operatorname{ch}L(\lambda)=\sum_{\mu\in P}(\dim L(\lambda)_\mu)e^\mu,
$$
where $L(\lambda)_\mu=\{v:Hv=\mu(H)v\text{ for all }H\in\mathfrak h\}$ is a <weight space>. For a weight $\nu$, define the <Weyl alternant> $A_\nu=\sum_{w\in W}\det(w)e^{w\nu}$; the sign $\det(w)=(-1)^{\ell(w)}$ is the determinant of $w$ on the real root span, equivalently the parity of its expression in simple reflections.
The <Weyl character formula> is
$$
\boxed{\operatorname{ch}L(\lambda)=\frac{A_{\lambda+\rho}}{A_\rho}
=\frac{\sum_{w\in W}\det(w)e^{w(\lambda+\rho)}}{e^\rho\prod_{\alpha\in R^+}(1-e^{-\alpha})}.}
$$
The second denominator is the <Weyl denominator formula>. The quotient can initially be taken in the fraction field of the group ring; the formula says it is the finite character on the left, so its apparent singularities are removable. The shifts by $\rho$ make $\lambda+\rho$ strictly dominant.
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