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Let $R^{+}$ be the positive roots, $W$ the Weyl group, $\ell (w)$ its Coxeter length, $\rho$ the half-sum of positive roots, and ${\alpha }^{\vee }$ a coroot. For a dominant integral highest weight $\lambda$, the Weyl character formula isTaking the value at the identity gives the Weyl dimension formula

For the q-character convention relevant to the Principal sl2 subalgebra, set $h_{\mathrm{pr}}=2{\rho }^{\vee }$, so ${\alpha }_i(h_{\mathrm{pr}})=2$ for every simple root, and defineThe q-character formula is the principal specialization of the Weyl character formula:

For the principal specialization, let $h_{\mathrm{pr}}=2{\rho }^{\vee }$, so every simple root takes value two on $h_{\mathrm{pr}}$. The q-character isIt is obtained from the Weyl character formula by substituting $e^{\mu }=q^{\mu (h_{\mathrm{pr}})}$.

For the complex special linear Lie algebra ${\mathfrak{sl}}_3$, tensoring an irreducible highest-weight representation by the defining representation gives the displayed sum, omitting terms with negative Dynkin labels. It follows by multiplying its Weyl character formula by $x_1+x_2+x_3$. In particular ${\Gamma }_{2,1}\otimes {\Gamma }_{1,0}={\Gamma }_{3,1}\oplus {\Gamma }_{1,2}\oplus {\Gamma }_{2,0}$, with dimensions $24+15+6=45$.

For the special linear Lie algebra $\mathfrak{sl}(3)$, with Dynkin labels indexing irreducible highest-weight representations,The summands have dimensions $60,24,15,15,6$, summing to $120=15\cdot 16/2$. The identity follows from the formal character of a weight module formula ${\chi }_{S^{2}W}(z)=({\chi }_W(z{)}^2+{\chi }_W(z^2))/2$ and the Weyl character formula.

In the symmetric power ${\mathrm{Sym}}^m{\mathbb{C}}^n$ of the defining unitary group representation, diagonal torus weights are indexed by nonnegative integer tuples summing to $m$, with one monomial per weight space. Torus averaging extracts a monomial from any nonzero invariant complex subspace. The complexified Lie algebra acts by $E_{ij}=x_i{\partial }_{x_j}$ on the symmetric algebra. Whenever $a_j>0$, the displayed action moves one unit of degree from $j$ to $i$ with nonzero coefficient. Finite transfers connect every degree-$m$ monomial, so the subspace is the entire representation. The case $m=0$ is the one-dimensional trivial representation. This proof needs neither the Weyl character formula nor the Weyl dimension formula.

For a Weyl group acting on a weight lattice, the displayed element of its group ring is alternating: applying $s\in W$ multiplies it by $\det (s)$, by reindexing the sum. It vanishes when $\nu$ is fixed by a root reflection. For the Weyl vector $\rho$, it is the Weyl denominator. The Weyl character formula is the ratio $A_{\lambda +\rho }/A_{\rho }$. These general root-system alternants specialize to monomial determinants in type A but are not restricted to that type.

The dimension obtained by evaluating the Weyl character formula at the identity is