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For a complex semisimple Lie algebra and an integer $k\ge 0$, the Weyl character formula and Weyl denominator formula giveApply the dilation $e^{\mu }\mapsto e^{(k+1)\mu }$ to the denominator identity to obtain the numerator. Cancellation leaves a finite geometric series for each positive root of a root system. Evaluating the resulting polynomial at all formal exponentials equal to one gives the dimension, without taking an undefined quotient at the identity.

For a root system with $N$ positive roots, put $A_{\nu }(H)={\sum }_{w\in W}\det (w)e^{(w\nu ,H)}$. Its degree-$m$ Taylor coefficient is alternating separately in $\nu$ and $H$, and hence divisible in each set of variables by the product of positive-root linear forms. Thus it vanishes for $m<N$ and its degree-$N$ term is $C{\prod }_{\alpha >0}(\nu ,\alpha ){\prod }_{\alpha >0}(H,\alpha )$. The Weyl denominator formula at $\nu =\rho$ determines the nonzero constant $C$. Taking the ratio $A_{\lambda +\rho }(tH)/A_{\rho }(tH)$ for regular $H$ and letting $t\to 0$ gives the Weyl dimension formula $\dim L(\lambda )={\prod }_{\alpha >0}(\lambda +\rho ,\alpha )/(\rho ,\alpha )$. Factors converting roots to coroots cancel.

Work over $\mathbb{C}$ with a finite-dimensional semisimple Lie algebra $\mathfrak{g}$. Choose a Cartan subalgebra $\mathfrak{h}$, its root system $R\subset {\mathfrak{h}}^{\ast }$, a positive system $R^{+}$ and its simple roots. Let $W$ be the Weyl group, generated by the corresponding root reflections, and put $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$, 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 iswhere $L(\lambda {)}_{\mu }=\{v:Hv=\mu (H)v\text{ for all }H\in \mathfrak{h}\}$ 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 isThe 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.

Let $N=|R^{+}|$ and identify the real weight and Cartan spaces using a Weyl group-invariant positive inner product. For a regular $H$, meaning $(\alpha ,H)\ne 0$ for every root, specialize $e^{\mu }$ to $e^{t(\mu ,H)}$. Then the value of the formal character at $t=0$ is $\dim L(\lambda )$, but both Weyl alternants in the Weyl character formula vanish there. We compute their leading terms instead of substituting into that zero-over-zero expression.

ExpandFor every root reflection $s$, replacing $\nu$ by $s\nu$ changes the sign of $P_m$, by replacing $w$ with $ws$ in the sum. Replacing $H$ by $sH$ also changes its sign, by replacing $w$ with $sw$. Thus the polynomial vanishes whenever $\nu$ or $H$ lies on any reflecting hyperplane. It is divisible by ${\prod }_{\alpha >0}(\nu ,\alpha )$ in its first variables and by ${\prod }_{\alpha >0}(H,\alpha )$ in its second variables. Distinct positive roots give distinct linear factors; each product has degree $N$.

The polynomial $P_m$ has degree $m$ in each group of variables. Therefore $P_m=0$ for $m<N$, and for $m=N$ it has the formwith a constant $C$ independent of both variables. At $\nu =\rho$, the Weyl denominator formula givesConsequently $C{\prod }_{\alpha >0}(\rho ,\alpha )=1$. For strictly dominant $\nu =\lambda +\rho$, its leading coefficient is nonzero, and the ratio of the two Weyl alternants tends toThe coroot is ${\alpha }^{\vee }=2\alpha /(\alpha ,\alpha )$, so the root-length factors cancel in each ratio. This proves the Weyl dimension formula by the lowest-degree alternant proof of the Weyl dimension formula, including the limiting step.

Apply the Weyl character formula with $\lambda =\rho$. In the Weyl denominator formula, replace every formal exponential $e^{\mu }$ by $e^{2\mu }$. This givesDividing by $A_{\rho }=e^{\rho }{\prod }_{\alpha >0}(1-e^{-\alpha })$ and cancelling each factor yieldsThe last expression is a compact way of writing the same character; individual half-root exponentials may need a larger formal lattice, but their full product lies in the group ring of the weight lattice. Expanding the first expression, a subset $S\subseteq R^{+}$ contributes the weight $\rho -{\sum }_{\alpha \in S}\alpha$. Different subsets with the same sum give that weight's multiplicity. In particular $\dim L(\rho )=2^{|R^{+}|}$, consistent with the Weyl dimension formula. This is the case $k=1$ of the character of a Weyl-vector multiple.

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, choose a Cartan subalgebra $\mathfrak{t}$ and a set $R^{+}$ of positive roots, and write $W$ for the Weyl group. Let $\lambda$ be a dominant integral weight and $L(\lambda )$ the finite-dimensional irreducible highest-weight representation with that highest weight. If $L(\lambda {)}_{\mu }$ denotes its weight space of weight $\mu$, its formal character is $\mathrm{ch}L(\lambda )={\sum }_{\mu }\dim L(\lambda {)}_{\mu }{e}^{\mu }$. The symbols $e^{\mu }$ belong to the group algebra of the weight lattice and multiply by $e^{\mu }{e}^{\nu }=e^{\mu +\nu }$; they are formal symbols, not matrix exponentials.

Put $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$, the half-sum of positive roots, and let $\ell (w)$ be the Coxeter length of $w$, the least number of simple reflections expressing it. Then $(-1{)}^{\ell (w)}=\det (w)$ in the real reflection representation. The Weyl character formula statesBy the Weyl denominator formula, the denominator is $e^{\rho }{\prod }_{\alpha \in R^{+}}(1-e^{-\alpha })$, giving the equivalent expressionThe quotient can be interpreted in the fraction field of the formal group algebra; the theorem says it simplifies to the finite formal character above. In particular the apparent denominator does not mean that the representation has infinitely many weights.

The Weyl group acts on the weight lattice by its root reflections and extends to ring automorphisms of the group ring of a weight lattice:Weights and their multiplicities in a finite-dimensional Lie algebra representation of a semisimple Lie algebra are permuted by this action, so its formal character is Weyl-invariant.

Choose a set ${\Phi }^{+}$ of positive roots and putHere ${\omega }_i$ are the fundamental weights, and $\det (w)=(-1{)}^{\ell (w)}$, where the Coxeter length $\ell (w)$ counts positive roots carried to negative ones. For a dominant integral weight $\lambda$, let $L(\lambda )$ denote the finite-dimensional Irreducible Lie algebra representation of highest weight $\lambda$. The Weyl character formula isThe second denominator expression is the Weyl denominator formula. Both alternants change sign under each root reflection; their quotient is invariant. Although written as a fraction, the theorem asserts that it is a finite element of $\mathbb{Z}[{\Lambda }_W]$ with the nonnegative coefficients given by weight multiplicities. The highest-weight classification specifies which $\lambda$ label the irreducibles, while the root system and choice of positive roots determine every other term.

Take an integer $k\ge 0$, so $k\rho$ is a dominant integral weight. Apply the dilation $e^{\mu }\mapsto e^{(k+1)\mu }$ to the Weyl denominator formula:This is exactly the numerator of the Weyl character formula for $L(k\rho )$. Divide by its denominator and use a finite geometric series in each root direction:This character of a Weyl-vector multiple also gives $\dim L(k\rho )=(k+1{)}^{|R^{+}|}$ by evaluating every formal exponential at one. This evaluation is safe in the final polynomial expression, without dividing by a zero denominator.

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.

For a dominant integral weight $\lambda$, the Weyl character formula iswhere $W$ is the Weyl group, $\ell$ is Coxeter length, and $\rho$ is the half-sum of positive roots. The Weyl denominator formula is

Set $\lambda =k\rho$. Apply the denominator identity after replacing every formal exponential $e^{\mu }$ by $e^{(k+1)\mu }$:Dividing this by the ordinary denominator gives

The Weyl denominator is the alternating expression in the Weyl denominator formula. Its product form uses the Weyl vector $\rho$, half the positive-root sum. On a compact maximal torus, the factor $e^{\rho }$ has unit modulus; if it is only defined on a covering torus, the squared modulus still descends. Thus $|\Delta (t){|}^2={\prod }_{\alpha >0}|1-\alpha (t){|}^2$ is a globally defined density in the Weyl integration formula. For the unitary group an integer central shift gives the Vandermonde determinant $\det (z_i^{n-j})$, whose modulus is the same.