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| In [[mathematics]], the '''Weyl character formula''' in [[representation theory]] describes the [[character (mathematics)|character]]s of irreducible representations of [[compact Lie group]]s in terms of their [[highest weight]]s. It was proved by {{harvs|txt|author-link=Hermann Weyl|first=Hermann |last=Weyl|year1=1925|year2=1926a|year3=1926b}}.
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| By definition, the character of a representation ''r'' of ''G'' is the [[trace of a matrix|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 representation]]s; ''irreducible'' therefore means the same as ''indecomposable'', i.e. not a direct sum of two subrepresentations.
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| ==Statement of Weyl character formula==
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| The character of an [[irreducible representation]] <math>V</math> of a complex semisimple Lie algebra <math>\mathfrak{g}</math> is given by
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| :<math>\operatorname{ch}(V) = \frac{\sum_{w\in W} \varepsilon(w) e^{w(\lambda+\rho)}}{e^{\rho}\prod_{\alpha \in \Delta^{+}}(1-e^{-\alpha})}</math>
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| where
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| *<math>W</math> is the [[Weyl group]];
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| *<math>\Delta^{+}</math> is the subset of the [[positive root]]s of the [[root system]] <math>\Delta</math>;
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| *<math>\rho</math> is half of the sum of the positive roots;
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| *<math>\lambda</math> is the [[highest weight]] of the irreducible representation <math>V</math>;
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| *<math>\varepsilon(w)</math> is the determinant of the action of <math>w</math> on <math>\mathfrak{h}</math>. This is equal to <math>(-1)^{\ell(w)}</math>, where <math>\ell(w)</math> is the [[Weyl group#Coxeter group structure|length of the Weyl group element]], defined to be the minimal number of reflections with respect to simple roots such that <math>w</math> equals the product of those reflections.
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| The character of an irreducible representation <math>V</math> of a compact connected Lie group <math>G</math> is given by
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| :<math>\operatorname{ch}(V) = \frac{\sum_{w\in W} \varepsilon(w) \xi_{w(\lambda+\rho)-\rho}}{\prod_{\alpha \in \Delta^{+}}(1-\xi_{-\alpha})}</math> | |
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| where <math>\xi_{\alpha}</math> is the character on <math>T</math> with differential <math>\alpha</math> on the Lie algebra <math>\mathfrak{t}_{0}</math> of the maximal Torus <math>T</math>.
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| If <math>\rho</math> is the differential of a character of <math>T</math>, e.g. if <math>G</math> is simply connected, this can be reformulated as
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| :<math>\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)}}</math>
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| ==Weyl denominator formula==
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| In the special case of the trivial 1 dimensional representation the character is 1, so the Weyl character formula becomes the '''Weyl denominator formula''':
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| :<math>{\sum_{w\in W} \varepsilon(w)e^{w(\rho)} = e^{\rho}\prod_{\alpha \in \Delta^{+}}(1-e^{-\alpha})}.</math>
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| For special unitary groups, this is equivalent to the expression
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| :<math>
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| \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) </math>
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| for the [[Vandermonde determinant]].
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| ==Weyl dimension formula==
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| By specialization to the trace of the identity element, Weyl's character formula gives the '''Weyl dimension formula'''
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| ::<math>\dim(V_\Lambda) = {\prod_{\alpha \in \Delta^{+}}(\Lambda+\rho,\alpha) \over \prod_{\alpha \in \Delta^{+}}(\rho,\alpha)}</math>
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| for the dimension
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| of a finite dimensional representation ''V''<sub>Λ</sub> 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
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| 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.
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| ==Freudenthal's formula==
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| [[Hans Freudenthal]]'s formula is a recursive formula for the weight multiplicities that is equivalent to the Weyl character formula, but is sometimes
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| easier to use for calculations as there can be far fewer terms to sum. It states
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| ::<math> (\|\Lambda+\rho\|^2 - \|\lambda+\rho\|^2)\dim V_\lambda
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| = 2 \sum_{\alpha \in \Delta^{+}}\sum_{j\ge 1} (\lambda+j\alpha, \alpha)\dim V_{\lambda+j\alpha}</math>
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| where
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| *Λ is a highest weight,
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| *λ is some other weight,
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| * dim V<sub>λ</sub> is the multiplicity of the weight λ
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| *ρ is the Weyl vector
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| *The first sum is over all positive roots α.
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| ==Weyl–Kac character formula==
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| The Weyl character formula also holds for integrable highest weight representations of [[Kac–Moody algebra]]s, when it is known as the '''Weyl–Kac character formula'''. Similarly there is a denominator identity for [[Kac–Moody algebra]]s, which in the case of the affine Lie algebras is equivalent to the '''[[Ian G. Macdonald|Macdonald]] identities'''. In the simplest case of the affine Lie algebra of type ''A''<sub>1</sub> this is the [[Jacobi triple product]] identity
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| :<math>\prod_{m=1}^\infty
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| \left( 1 - x^{2m}\right)
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| \left( 1 - x^{2m-1} y\right)
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| \left( 1 - x^{2m-1} y^{-1}\right)
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| = \sum_{n=-\infty}^\infty (-1)^n x^{n^2} y^n.
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| </math>
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| The character formula can also be extended to integrable highest weight representations of [[generalized Kac–Moody algebra]]s, when the character is given by
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| :<math>{\sum_{w\in W} (-1)^{\ell(w)}w(e^{\lambda+\rho}S) \over e^{\rho}\prod_{\alpha \in \Delta^{+}}(1-e^{-\alpha})}.</math>
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| Here ''S'' is a correction term given in terms of the imaginary simple roots by
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| :<math> S=\sum_I (-1)^{|I|}e^{\Sigma I} \, </math>
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| 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 Σ''I'' is the sum of the elements of ''I''.
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| The denominator formula for the [[monster Lie algebra]] is the product formula
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| ::<math>j(p)-j(q) = \left({1 \over p} - {1 \over q}\right) \prod_{n,m=1}^\infty (1-p^n q^m)^{c_{nm}}</math>
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| for the [[elliptic modular function]] ''j''.
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| 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:
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| ::<math> (\beta,\beta-2\rho)c_\beta = \sum_{\gamma+\delta=\beta} (\gamma,\delta)c_\gamma c_\delta \, </math>
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| where the sum is over positive roots γ, δ, and
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| ::<math> c_\beta = \sum_{n\ge 1} {\operatorname{mult}(\beta/n)\over n}.</math>
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| ==Harish-Chandra Character Formula==
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| Harish-Chandra showed that Weyl's character formula admits a generalization to representations of a real, [[reductive group]]. Suppose <math> \pi </math> is an irreducible, [[admissible representation]] of a real, reductive group G with [[infinitesimal character]] <math> \lambda </math>. Let <math> \Theta_{\pi} </math> be the [[Harish-Chandra character]] of <math> \pi </math>; 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
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| ::<math> \Theta_{\pi}|_{H'}= {\sum_{w\in W/W_{\lambda}} a_w e^{w\lambda} \over e^{\rho}\prod_{\alpha \in \Delta^{+}}(1-e^{-\alpha})}.</math>
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| Here
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| * W is the complex Weyl group of <math> H_{\mathbb{C}} </math> with respect to <math> G_{\mathbb{C}} </math>
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| * <math> W_{\lambda} </math> is the stabilizer of <math> \lambda </math> in W
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| and the rest of the notation is as above.
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| The coefficients <math> a_w </math> are still not well understood. Results on these coefficients may be found in papers of Herb, Adams, Schmid, and Schmid-Vilonen among others.
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| == See also ==
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| *[[Algebraic character]]
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| *[[Demazure character formula]]
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| ==References==
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| *''Infinite dimensional Lie algebras'', V. G. Kac, ISBN 0-521-37215-1
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| *{{springer|id=W/w130070|title=Weyl–Kac character formula|author=Duncan J. Melville}}
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| *{{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. I | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF01506234 | year=1925 | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=23 | pages=271–309}}
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| *{{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. II | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF01216788 | year=1926a | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=24 | pages=328–376}}
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| *{{Citation | last1=Weyl | first1=Hermann | author1-link=Hermann Weyl | title=Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen. III | publisher=Springer Berlin / Heidelberg | doi=10.1007/BF01216789 | year=1926b | journal=[[Mathematische Zeitschrift]] | issn=0025-5874 | volume=24 | pages=377–395}}
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| [[Category:Representation theory of Lie groups]]
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