Weyl character formula

For a dominant integral weight $\lambda$, the irreducible character is

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.

For strictly decreasing integer tuples $\kappa ,\eta$, let $A_{\kappa }(z)=\det (z_i^{{\kappa }_j})$ on the diagonal maximal torus of the unitary group. Expanding both determinants, Fourier orthogonality makes the integral zero unless the exponent tuples have the same permutation orbit. Strict decreasing order makes that equivalent to $\kappa =\eta$; the $n!$ matching permutation terms then each contribute one. An alternating Laurent polynomial is a finite linear combination of these alternants. For an irreducible character ${\chi }_{\lambda }$, its highest monomial gives coefficient one for $A_{\lambda +\delta }$ in ${\chi }_{\lambda }{A}_{\delta }$, with $\delta =(n-1,\dots ,0)$. The Weyl integration formula and Schur orthogonality relations force the sum of squared coefficients to be one, so all other coefficients vanish. This proves the unitary Weyl character formula directly.

For a dominant integer tuple $\lambda$ and $\delta =(m-1,\dots ,0)$, the rational Schur character is $\det (x_i^{{\lambda }_j+m-j})/\det (x_i^{m-j})$. For partitions it is a symmetric polynomial; for general tuples it is Laurent on the diagonal torus. The formula follows by comparing the trace of a permuted tensor power with the Frobenius alternant character formula and using independence of symmetric-group characters. Apparent poles at coincident nonzero eigenvalues are removable.

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 positive root system ${\Phi }^{+}$,

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.

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

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.

For the G2 root system with ${\alpha }_1$ short and ${\alpha }_2$ long, the fundamental weights are ${\omega }_1=2{\alpha }_1+{\alpha }_2$ and ${\omega }_2=3{\alpha }_1+2{\alpha }_2$. The positive coroots, relative to the simple coroots, have coordinates $(1,0),(0,1),(1,3),(2,3),(1,1),(1,2)$. The Weyl dimension formula therefore gives, for nonnegative integers $a,b$,The denominator is the product of the pairings of the Weyl vector with those coroots. The dimensions at ${\omega }_1,{\omega }_2,2{\omega }_1$ are seven, fourteen and twenty-seven.

For the C2 root system with its first simple root short, the irreducible highest-weight representation of highest weight $a{\omega }_1+b{\omega }_2$, where $a,b\in {\mathbb{Z}}_{\ge 0}$, has dimension

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 $e^{\mu }=q^{\mu (h_{\mathrm{pr}})}$.

Use ${\mathrm{ch}}_{q}V={\mathrm{tr}}_V{q}^{2{\rho }^{\vee }}$. The positive roots of a root system of the G2 root system have simple-root heights $1,1,2,3,4,5$. Each root space has dimension one, and the zero weight space has dimension two. ConsequentlyBy the classification of finite-dimensional sl2 representations, restriction to the Principal sl2 subalgebra is $L_{10}\oplus L_2$, of dimensions eleven and three. If the grading uses ${\rho }^{\vee }$ instead, divide every exponent by two.