I paraphrase part of the wikipedia article on the Weyl character formula: Weyl character formula.
If $\pi$ is an irreducible finite-dimensional representation of a complex semisimple Lie algebra $\mathfrak{g}$ and $\mathfrak{h}$ is a choice of Cartan subalgebra of $\mathfrak{g}$, then the Weyl character formula states that the character $\operatorname{ch}_\pi$ of $\pi$ is given by
$$ \operatorname{ch}_\pi(H) = \frac{\sum_{w \in W} \epsilon(w) e^{w(\lambda+\rho)(H)}}{\prod_{\alpha \in \Delta^+}(e^{\alpha(H)/2} – e^{-\alpha(H)/2})},$$
where $W$ is the Weyl group, $H \in \mathfrak{h}$, $\epsilon(w)$ is the determinant of the action of $w \in W$ on the Cartan subalgebra $\mathfrak{h}$, $\Delta^+$ denotes the set of positive roots of $(\mathfrak{g}, \mathfrak{h})$, $\lambda$ denotes the highest weight of $\pi$ and $\rho$ is half the sum of all positive roots of $(\mathfrak{g}, \mathfrak{h})$ (i.e. half the sum of all the elements of $\Delta^+$).
If $\mathfrak{g} = \mathfrak{sl}(n)$, it is known that the denominator of the RHS of the Weyl character formula can be written as a determinant (actually a Vandermonde determinant). While it does seem counterproductive, since the numerator looks simple enough (well, to some extent), yet I am interested whether or not the numerator can also be written as a determinant, at least for $\mathfrak{g} = \mathfrak{sl}(n)$, though I am also interested in the other cases too. After all, the Weyl group in this special case is $S_n$, the symmetric group on $n$ elements, and an $n \times n$ determinant can be expanded as an alternating sum over $S_n$. So this does seem promising. I apologize if it turns out to be a trivial question perhaps (I currently have limited access to online journals etc.).
Best Answer
The classical definition of the Schur polynomials, which considerably predates the Weyl character formula, is as a ratio of two determinants (a so-called "bialternant"): see, e.g., https://en.wikipedia.org/wiki/Schur_polynomial#Definition_(Jacobi's_bialternant_formula).
Of course the Schur polynomials are $\mathfrak{sl}_n$ characters. There are similar things in other types too, if you are interested in that (for example, see Proposition 1.1 in Okada's "Applications of Minor Summation Formulas to Rectangular-Shaped Representations of Classical Groups", https://doi.org/10.1006/jabr.1997.7408 ).