Weyl character formula

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 is named after Hermann 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 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.

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})}


*ρ is the Weyl vector of the group "G", defined to be half the sum of the positive roots;
*"W" is the Weyl group;
*λ is the highest weight of the irreducible representation "V";
*α runs over the positive roots 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 ifor the Vandermonde 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}


*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

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} 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 algebras, 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


*"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

Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • List of topics named after Hermann Weyl — This is a list of topics named after Hermann Weyl, the influential German mathematician. * Weyl algebra * Weyl basis of the gamma matrices * Weyl chamber * Weyl character formula * Weyl s criterion * Weyl curvature: see Weyl tensor * Weyl… …   Wikipedia

  • Hermann Weyl — Pour les articles homonymes, voir Weil et Veil.  En particulier, ne pas confondre avec cet autre mathématicien : André Weil …   Wikipédia en Français

  • Algebraic character — is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite dimensional representation and is analogous to the Harish Chandra character of the representations of… …   Wikipedia

  • Hermann Weyl — Infobox Scientist box width = name = Hermann Klaus Hugo Weyl image size = caption = Hermann Weyl (left) and Ernst Peschl (right) birth date = Birth date|1885|11|9|df=y birth place = Elmshorn, Germany death date = Death date and… …   Wikipedia

  • Smith–Minkowski–Siegel mass formula — In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism… …   Wikipedia

  • Demazure module — In mathematics, a Demazure module, introduced by Demazure (1974a, 1974b). is a submodule of a finite dimensional representation generated by an extremal weight space under the action of a Borel subalgebra. The Demazure character formula,… …   Wikipedia

  • Zonal spherical function — In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K invariant vector in an… …   Wikipedia

  • Glossary of semisimple groups — This is a glossary for the terminology applied in the mathematical theories of semisimple Lie groups. It also covers terms related to their Lie algebras, their representation theory, and various geometric, algebraic and combinatorial structures… …   Wikipedia

  • Discrete series representation — In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²(G). In the Plancherel measure, such… …   Wikipedia

  • Kazhdan–Lusztig polynomial — In representation theory, a Kazhdan–Lusztig polynomial P y,w ( q ) is a member of a family of integral polynomials introduced in work of David Kazhdan and George Lusztig Harv|Kazhdan|Lusztig|1979. They are indexed by pairs of elements y , w of a… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”