Weyl character formula

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

The character of an irreducible representation $V$ of a complex semisimple Lie algebra ${\mathfrak {g}}$ is given by^[1]

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

where

$W$ is the Weyl group;
$\Delta ^{+}$ 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 the Cartan subalgebra ${\mathfrak {h}}\subset {\mathfrak {g}}$ . 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.

Using the Weyl denominator formula, the character formula may be rewritten as

\operatorname {ch} (V){\sum _{w\in W}\varepsilon (w)e^{w(\rho )}}=\sum _{w\in W}\varepsilon (w)e^{w(\lambda +\rho )}

Note that the character is itself a large sum of exponentials. We then multiply the character by an alternating sum of exponentials. The surprising part of the character formula is that when we compute this product, only a small number of terms actually remain. Many more terms than this occur at least once in the product of the character and the Weyl denominator, but most of these terms cancel out to zero.^[2] The only terms that survive are the terms that occur only once, namely $e^{\lambda +\rho }$ (which is obtained by taking the highest weight from $\operatorname {ch} (V)$ and the highest weight from the Weyl denominator) and things in the Weyl-group orbit of $e^{\lambda +\rho }$ .

The character of an irreducible representation $V$ of a compact connected Lie group $G$ is given by^[3]

\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,^[4] 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

In the special case of the trivial 1-dimensional representation the character is 1, so the Weyl character formula becomes the Weyl denominator formula:^[5]

{\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}}\operatorname {sgn}(\sigma )\,X_{1}^{\sigma (1)-1}\cdots X_{n}^{\sigma (n)-1}=\prod _{1\leq i<j\leq n}(X_{j}-X_{i})

for the Vandermonde determinant.^[6]

Weyl dimension formula

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 V_Λ 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.^[7]

Freudenthal's formula

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})m_{\Lambda }(\lambda )=2\sum _{\alpha \in \Delta ^{+}}\sum _{j\geq 1}(\lambda +j\alpha ,\alpha )m_{\Lambda }(\lambda +j\alpha )

where

Λ is a highest weight,
λ is some other weight,
m_Λ(λ) is the multiplicity of the weight λ in the irreducible representation V_Λ
ρ is the Weyl vector
The first sum is over all positive roots α.

Weyl–Kac character formula

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\geq 1}{\operatorname {mult} (\beta /n) \over n}.

Harish-Chandra Character Formula

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.

References

↑ Hall 2015 Theorem 10.14.
↑ Hall 2015 Section 10.4.
↑ Hall 2015 Section 12.4
↑ Hall 2015 Corollary 13.20
↑ Hall 2015 Lemma 10.28.
↑ Hall 2015 Exercise 9 in Chapter 10.
↑ Hall 2015 Section 10.5.

Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer
Infinite dimensional Lie algebras, V. G. Kac, ISBN 0-521-37215-1
Duncan J. Melville (2001), "Weyl–Kac character formula", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Weyl, Hermann (1925), "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. I", Mathematische Zeitschrift, Springer Berlin / Heidelberg, 23: 271–309, doi:10.1007/BF01506234, ISSN 0025-5874
Weyl, Hermann (1926a), "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. II", Mathematische Zeitschrift, Springer Berlin / Heidelberg, 24: 328–376, doi:10.1007/BF01216788, ISSN 0025-5874
Weyl, Hermann (1926b), "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. III", Mathematische Zeitschrift, Springer Berlin / Heidelberg, 24: 377–395, doi:10.1007/BF01216789, ISSN 0025-5874

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