- Weyl character formula
In

mathematics , the**Weyl character formula**inrepresentation theory describes the characters of irreducible representations ofcompact Lie group s in terms of theirhighest weight s. It is named afterHermann Weyl , who proved it in the late 1920s.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 aclosed formula for the χ, in terms of other objects constructed from "G" and itsLie algebra . The representations in question here are complex, and so without loss of generality areunitary representation s; "irreducible" therefore means the same as "indecomposable", i.e. not a direct sum of two subrepresentations.**tatement of Weyl character formula**The character of an

irreducible representation "V" of a compact Lie group "G" is given by:$ch(V)=\{sum\_\{win\; W\}\; (-1)^\{ell(w)\}w(e^\{lambda+\; ho\})\; over\; e^\{\; ho\}prod\_\{alpha>0\}(1-e^\{-alpha\})\}$

where

*ρ is the Weyl vector of the group "G", defined to be half the sum of the positive roots;

*"W" is theWeyl group ;

*λ is thehighest weight of the irreducible representation "V";

*α runs over thepositive root s of the Lie group.**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**::$\{sum\_\{win\; W\}\; (-1)^\{ell(w)\}w(e^\{\; ho\})\; =\; e^\{\; ho\}prod\_\{alpha>0\}(1-e^\{-alpha\})\}.$

For special unitary groups, this is equivalent to the expression :$sum\_\{sigma\; in\; S\_n\}\; sgn(sigma)\; ,\; alpha\_1^\{sigma(1)-1\}\; cdots\; alpha\_n^\{sigma(n)-1\}\; =prod\_\{1le\; i\}\; (alpha\_j-alpha\_i)\; math>for\; theVandermonde\; determinant.$

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

**Weyl dimension formula**::$dim(V\_Lambda)\; =\; \{prod\_\{alpha>0\}(Lambda+\; ho,alpha)\; over\; prod\_\{alpha>0\}(\; ho,alpha)\}$for the dimensionof 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 boththe 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**Hans Freudenthal 's formula is a recursive formula for the weight multiplicities that is equivalent to the Weyl character formula, but is sometimeseasier to use for calculations as there can be far fewer terms to sum. It states::$((Lambda+\; ho)^2\; -\; (lambda+\; ho)^2)dim\; V\_lambda=\; 2\; sum\_\{alpha>0\}sum\_\{jge\; 1\}\; (lambda+jalpha,\; alpha)dim\; V\_\{lambda+jalpha\}$

where

*Λ is a highest weight,

*λ is some other weight,

* dim V_{λ}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**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 forKac-Moody algebra s, 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 A_{1}this is theJacobi triple product identity:$prod\_\{m=1\}^infty\; left(\; 1\; -\; x^\{2m\}\; ight)left(\; 1\; -\; x^\{2m-1\}\; y\; ight)left(\; 1\; -\; x^\{2m-1\}\; y^\{-1\}\; ight)=\; sum\_\{n=-infty\}^infty\; (-1)^n\; x^\{n^2\}\; y^\{n\}.$

The character formula can also be extended to integrable highest weight representations of

generalized Kac-Moody algebra s, when the character is given by:$\{sum\_\{win\; W\}\; (-1)^\{ell(w)\}w(e^\{lambda+\; ho\}S)\; over\; e^\{\; ho\}prod\_\{alpha>0\}(1-e^\{-alpha\})\}.$

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

:$left.S=sum\_\{I\}(-1)^e^\{Sigma\; I\}\; ight.$

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".

The denominator formula for the

monster Lie algebra is the product formula::$j(p)-j(q)\; =\; left(\{1\; over\; p\}\; -\; \{1\; over\; q\}\; ight)\; prod\_\{n,m=1\}^\{infty\}(1-p^n\; q^m)^\{c\_\{nm$

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:::$left.(eta,eta-2\; ho)c\_eta\; =\; sum\_\{gamma+delta=eta\}\; (gamma,delta)c\_gamma\; c\_delta\; ight.$where the sum is over positive roots γ, δ, and ::$c\_eta\; =\; sum\_\{nge\; 1\}$ m mult}(eta/n)over n}

**See also***

Algebraic character **References***"Infinite dimensional Lie algebras", V. G. Kac, ISBN 0-521-37215-1

*springer|id=W/w130070|title=Weyl–Kac character formula|author=Duncan J. Melville

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