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The formal character of a weight module for a minuscule representation of highest weight $\lambda$ is $\mathrm{ch}V(\lambda )={\sum }_{\mu \in W\lambda }{e}^{\mu }$. The sum is over distinct points of the Weyl group orbit, so $\dim V(\lambda )=|W\lambda |$.

Use the generators $e,f,h$ of the sl2 Lie algebra with $[h,e]=2e$, $[h,f]=-2f$, and $[e,f]=h$. For a finite-dimensional Lie algebra representation, $h$ is diagonalizable with integral weights. Its formal character of a weight module isHere $q$ is a formal variable; equivalently the expression is ${\mathrm{tr}}_V(q^h)$, or the formal sum ${\sum }_m(\dim V_m)e^m$ with $e^m=q^m$. This records every weight multiplicity, rather than only the dimension.

The classification of finite-dimensional sl2 representations says that $L_d$, the irreducible of highest weight $d\ge 0$, has the weight string $d,d-2,\dots ,-d$, each with multiplicity one. HenceThe quotient denotes the displayed Laurent polynomial and has the removable value $d+1$ at $q=1$. Formal characters are additive on direct sums, so these formulas determine the character of every finite-dimensional module.

Choose a positive root system $R^{+}$ for a complex semisimple Lie algebra, with Weyl group $W$ and Weyl vector $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$. For a dominant integral weight $\lambda$, let $L(\lambda )$ denote its finite-dimensional irreducible representation, and use formal symbols with $e^{\mu }{e}^{\nu }=e^{\mu +\nu }$. Its formal character of a weight module is $\mathrm{ch}L(\lambda )={\sum }_{\mu }\dim L(\lambda {)}_{\mu }{e}^{\mu }$. The Weyl character formula saysHere $\det (w)=(-1{)}^{\ell (w)}$, the sign of the product of simple root reflections. The Weyl denominator formula isThe quotient is initially interpreted in the fraction field of the group algebra of the weight lattice; the theorem makes it a finite character.

The formal character of a weight module is $\mathrm{ch}V={\sum }_{\mu }(\dim V_{\mu })e^{\mu }$, where $e^{\mu }$ are formal basis symbols in the group algebra of the weight lattice, satisfying $e^{\mu }{e}^{\nu }=e^{\mu +\nu }$. For a finite-dimensional irreducible highest-weight representation, the highest weight has weight multiplicity one. Another standard result is that weight multiplicities are invariant under the Weyl group: $\dim V_{w\mu }=\dim V_{\mu }$.

A minuscule representation has all its weights in a single Weyl group orbit. Since the highest weight $\lambda$ is present, that orbit is $W\lambda$, and the invariance just stated shows every point of it is present with multiplicity one. Consequently the formal character of a minuscule representation isThe sum is over distinct weights, rather than over all elements of $W$: summing $e^{w\lambda }$ over $w\in W$ would count each weight $|{\mathrm{Stab}}_W(\lambda )|$ times.

For a dominant integral weight $\lambda$, the Weyl character formula isHere $W$ is the Weyl group, $\ell (w)$ its Coxeter length, $\rho$ the Weyl vector, and $\mathrm{ch}$ the formal character of a weight module. Taking the limit at the identity gives the Weyl dimension formula

Choose the Borel subalgebra $\mathfrak{b}=\mathfrak{t}\oplus {\mathfrak{n}}^{+}$ determined by the positive roots. Regard the one-dimensional space ${\mathbb{C}}_{\lambda }$ as a $\mathfrak{b}$-module on which ${\mathfrak{n}}^{+}$ acts by zero and $h\in \mathfrak{t}$ acts by $\lambda (h)$. The Verma module isThe Poincaré-Birkhoff-Witt theorem identifies it as a vector space with $U({\mathfrak{n}}^{-})$ acting on a highest-weight vector $v_{\lambda }$.

For each positive root $\alpha$, arbitrary powers of a negative-root vector contribute the geometric series $1+e^{-\alpha }+e^{-2\alpha }+\cdots$. Consequently the formal character of a weight module isThis product is interpreted in the completion of the group algebra in the negative-root direction; the PBW basis proves that every coefficient is the correct finite weight space dimension.

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.