Weyl character formula

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In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by Hermann Weyl (1925, 1926a, 1926b).

By definition, the character of a representation r of G is the trace of r(g), as a function of a group element g in G. The irreducible representations in this case are all finite-dimensional (this is part of the Peter–Weyl theorem); so the notion of trace is the usual one from linear algebra. Knowledge of the character χ of r is a good substitute for r itself, and can have algorithmic content. Weyl's formula is a closed formula for the χ, in terms of other objects constructed from G and its Lie algebra. The representations in question here are complex, and so without loss of generality are unitary representations; irreducible therefore means the same as indecomposable, i.e. not a direct sum of two subrepresentations.

Statement of Weyl character formula[edit]

The character of an irreducible representation V of a complex semisimple Lie algebra \mathfrak{g} is given by

\operatorname{ch}(V) = \frac{\sum_{w\in W} \varepsilon(w) e^{w(\lambda+\rho)}}{e^{\rho}\prod_{\alpha \in \Delta^{+}}(1-e^{-\alpha})}

where

  • \Delta^{+} is the subset of the positive roots of the root system \Delta;
  • \rho is the half sum of the positive roots;
  • \lambda is the highest weight of the irreducible representation V;
  • \varepsilon(w) is the determinant of the action of w on \mathfrak{h}. This is equal to (-1)^{\ell(w)}, where \ell(w) is the length of the Weyl group element, defined to be the minimal number of reflections with respect to simple roots such that w equals the product of those reflections.

The character of an irreducible representation V of a compact connected Lie group G is given by

\operatorname{ch}(V) = \frac{\sum_{w\in W} \varepsilon(w) \xi_{w(\lambda+\rho)-\rho}}{\prod_{\alpha \in \Delta^{+}}(1-\xi_{-\alpha})}

where \xi_{\alpha} is the character on T with differential \alpha on the Lie algebra \mathfrak{t}_{0} of the maximal Torus T.

If \rho is the differential of a character of T, e.g. if G is simply connected, this can be reformulated as

\operatorname{ch}(V) = \frac{\sum_{w\in W} \varepsilon(w) \xi_{w(\lambda+\rho)}}{\xi_{\rho}\prod_{\alpha \in \Delta^{+}}(1-\xi_{-\alpha})} = \frac{\sum_{w\in W} \varepsilon(w) \xi_{w(\lambda+\rho)}}{\sum_{w\in W} \varepsilon(w) \xi_{w(\rho)}}

Weyl denominator formula[edit]

In the special case of the trivial 1-dimensional representation the character is 1, so the Weyl character formula becomes the Weyl denominator formula:

{\sum_{w\in W} \varepsilon(w)e^{w(\rho)} = e^{\rho}\prod_{\alpha \in \Delta^{+}}(1-e^{-\alpha})}.

For special unitary groups, this is equivalent to the expression

 
\sum_{\sigma \in S_n} \sgn(\sigma) \, X_1^{\sigma(1)-1} \cdots X_n^{\sigma(n)-1} =\prod_{1\le i<j\le n} (X_j-X_i)

for the Vandermonde determinant.

Weyl dimension formula[edit]

By specialization to the trace of the identity element, Weyl's character formula gives the Weyl dimension formula

\dim(V_\Lambda) = {\prod_{\alpha \in \Delta^{+}}(\Lambda+\rho,\alpha) \over \prod_{\alpha \in \Delta^{+}}(\rho,\alpha)}

for the dimension of a finite dimensional representation VΛ with highest weight Λ. (As usual, ρ is the Weyl vector and the products run over positive roots α.) The specialization is not completely trivial, because both the numerator and denominator of the Weyl character formula vanish to high order at the identity element, so it is necessary to take a limit of the trace of an element tending to the identity.

Freudenthal's formula[edit]

Hans Freudenthal's formula is a recursive formula for the weight multiplicities that is equivalent to the Weyl character formula, but is sometimes easier to use for calculations as there can be far fewer terms to sum. It states

 (\|\Lambda+\rho\|^2 - \|\lambda+\rho\|^2)\dim V_\lambda
= 2 \sum_{\alpha \in \Delta^{+}}\sum_{j\ge 1} (\lambda+j\alpha, \alpha)\dim V_{\lambda+j\alpha}

where

  • Λ is a highest weight,
  • λ is some other weight,
  • dim Vλ is the multiplicity of the weight λ
  • ρ is the Weyl vector
  • The first sum is over all positive roots α.

Weyl–Kac character formula[edit]

The Weyl character formula also holds for integrable highest weight representations of Kac–Moody algebras, when it is known as the Weyl–Kac character formula. Similarly there is a denominator identity for Kac–Moody algebras, which in the case of the affine Lie algebras is equivalent to the Macdonald identities. In the simplest case of the affine Lie algebra of type A1 this is the Jacobi triple product identity

\prod_{m=1}^\infty 
\left( 1 - x^{2m}\right)
\left( 1 - x^{2m-1} y\right)
\left( 1 - x^{2m-1} y^{-1}\right)
= \sum_{n=-\infty}^\infty (-1)^n x^{n^2} y^n.

The character formula can also be extended to integrable highest weight representations of generalized Kac–Moody algebras, when the character is given by

{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho}S) \over e^{\rho}\prod_{\alpha \in \Delta^{+}}(1-e^{-\alpha})}.

Here S is a correction term given in terms of the imaginary simple roots by

 S=\sum_I (-1)^{|I|}e^{\Sigma I} \,

where the sum runs over all finite subsets I of the imaginary simple roots which are pairwise orthogonal and orthogonal to the highest weight λ, and |I| is the cardinality of I and ΣI is the sum of the elements of I.

The denominator formula for the monster Lie algebra is the product formula

j(p)-j(q) = \left({1 \over p} - {1 \over q}\right) \prod_{n,m=1}^\infty (1-p^n q^m)^{c_{nm}}

for the elliptic modular function j.

Peterson gave a recursion formula for the multiplicities mult(β) of the roots β of a symmetrizable (generalized) Kac–Moody algebra, which is equivalent to the Weyl–Kac denominator formula, but easier to use for calculations:

 (\beta,\beta-2\rho)c_\beta = \sum_{\gamma+\delta=\beta} (\gamma,\delta)c_\gamma c_\delta \,

where the sum is over positive roots γ, δ, and

 c_\beta = \sum_{n\ge 1} {\operatorname{mult}(\beta/n)\over n}.

Harish-Chandra Character Formula[edit]

Harish-Chandra showed that Weyl's character formula admits a generalization to representations of a real, reductive group. Suppose  \pi is an irreducible, admissible representation of a real, reductive group G with infinitesimal character  \lambda . Let  \Theta_{\pi} be the Harish-Chandra character of  \pi ; it is given by integration against an analytic function on the regular set. If H is a Cartan subgroup of G and H' is the set of regular elements in H, then

 \Theta_{\pi}|_{H'}= {\sum_{w\in W/W_{\lambda}} a_w e^{w\lambda} \over e^{\rho}\prod_{\alpha \in \Delta^{+}}(1-e^{-\alpha})}.

Here

  • W is the complex Weyl group of  H_{\mathbb{C}} with respect to  G_{\mathbb{C}}
  •  W_{\lambda} is the stabilizer of  \lambda in W

and the rest of the notation is as above.

The coefficients  a_w are still not well understood. Results on these coefficients may be found in papers of Herb, Adams, Schmid, and Schmid-Vilonen among others.

See also[edit]

References[edit]