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.^[1] It was proved by Hermann Weyl (1925, 1926a, 1926b). There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra.^[2] In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation.^[3] Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula.

By definition, the character χdisplaystyle chi of a representation πdisplaystyle pi of G is the trace of π(g)displaystyle pi (g), as a function of a group element g∈Gdisplaystyle gin 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 χdisplaystyle chi of πdisplaystyle pi gives a lot of information about πdisplaystyle pi itself.

Weyl's formula is a closed formula for the character χdisplaystyle chi , in terms of other objects constructed from G and its Lie algebra.

Statement of Weyl character formula[edit]

The character formula can be expressed in terms of representations of complex semisimple Lie algebras or in terms of the (essentially equivalent) representation theory of compact Lie groups.

Complex semisimple Lie algebras[edit]

Let πdisplaystyle pi be an irreducible, finite-dimensional representation of a complex semisimple Lie algebra gdisplaystyle mathfrak g. Suppose hdisplaystyle mathfrak h is a Cartan subalgebra of gdisplaystyle mathfrak g. The character of πdisplaystyle pi is then the function chπ:h→Cdisplaystyle mathrm ch _pi :mathfrak hrightarrow mathbb C defined by

chπ(H)=trace(eπ(H))displaystyle mathrm ch _pi (H)=mathrm trace (e^pi (H)).

The value of the character at H=0displaystyle H=0 is the dimension of πdisplaystyle pi . By elementary considerations, the character may be computed as

chπ(H)=∑μmμeμ(H)displaystyle mathrm ch _pi (H)=sum _mu m_mu e^mu (H),

where the sum ranges over all the weights μdisplaystyle mu of πdisplaystyle pi and where mμdisplaystyle m_mu is the multiplicity of μdisplaystyle mu . (The preceding expression is sometimes taken as the definition of the character.)

The character formula states^[4] that chπ(H)displaystyle mathrm ch _pi (H) may also be computed as

chπ(H)=∑w∈Wε(w)ew(λ+ρ)(H)∏α∈Δ+(eα(H)/2−e−α(H)/2)displaystyle mathrm ch _pi (H)=frac sum _win Wvarepsilon (w)e^w(lambda +rho )(H)prod _alpha in Delta ^+(e^alpha (H)/2-e^-alpha (H)/2)

where

• 
  Wdisplaystyle W is the Weyl group;


• 
  Δ+displaystyle Delta ^+ is the set of the positive roots of the root system Δdisplaystyle Delta ;


• 
  ρdisplaystyle rho is the half-sum of the positive roots;


• 
  λdisplaystyle lambda is the highest weight of the irreducible representation Vdisplaystyle V;


• 
  ε(w)displaystyle varepsilon (w) is the determinant of the action of wdisplaystyle w on the Cartan subalgebra h⊂gdisplaystyle mathfrak hsubset mathfrak g. This is equal to (−1)ℓ(w)displaystyle (-1)^ell (w), where ℓ(w)displaystyle 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 wdisplaystyle w equals the product of those reflections.

Discussion[edit]

Using the Weyl denominator formula (described below), the character formula may be rewritten as

chπ(H)=∑w∈Wε(w)ew(λ+ρ)(H)∑w∈Wε(w)ew(ρ)(H)displaystyle mathrm ch _pi (H)=frac sum _win Wvarepsilon (w)e^w(lambda +rho )(H)sum _win Wvarepsilon (w)e^w(rho )(H),

or, equivalently,

chπ(H)∑w∈Wε(w)ew(ρ)(H)=∑w∈Wε(w)ew(λ+ρ)(H)displaystyle mathrm ch _pi (H)sum _win Wvarepsilon (w)e^w(rho )(H)=sum _win Wvarepsilon (w)e^w(lambda +rho )(H).

Note that the character is itself a large sum of exponentials. In this last expression, we then multiply the character by an alternating sum of exponentials—which seemingly will result in an even larger 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.^[5] The only terms that survive are the terms that occur only once, namely e(λ+ρ)(H)displaystyle e^(lambda +rho )(H) (which is obtained by taking the highest weight from chπdisplaystyle mathrm ch _pi and the highest weight from the Weyl denominator) and things in the Weyl-group orbit of e(λ+ρ)(H)displaystyle e^(lambda +rho )(H).

Compact Lie groups[edit]

Let Kdisplaystyle K be a compact, connected Lie group and let Tdisplaystyle T be a maximal torus in Kdisplaystyle K. Let Πdisplaystyle Pi be an irreducible representation of Kdisplaystyle K. Then we define the character of Πdisplaystyle Pi to be the function

X(x)=trace(Π(x)),x∈Kdisplaystyle mathrm X (x)=mathrm trace (Pi (x)),quad xin K.

The character is easily seen to be a class function on Kdisplaystyle K and the Peter–Weyl theorem asserts that the characters form an orthonormal basis for the space of square-integrable class functions on Kdisplaystyle K.^[6]

Since Xdisplaystyle mathrm X is a class function, it is determined by its restriction to Tdisplaystyle T. Now, for Hdisplaystyle H in the Lie algebra tdisplaystyle mathfrak t of Tdisplaystyle T, we have

trace(Π(eH))=trace(eπ(H))displaystyle mathrm trace (Pi (e^H))=mathrm trace (e^pi (H)),

where πdisplaystyle pi is the associated representation of the Lie algebra kdisplaystyle mathfrak k of Kdisplaystyle K. Thus, the function H↦trace(Π(eH))displaystyle Hmapsto mathrm trace (Pi (e^H)) is simply the character of the associated representation πdisplaystyle pi of kdisplaystyle mathfrak k, as described in the previous subsection. The restriction of the character of Πdisplaystyle Pi to Tdisplaystyle T is then given by the same formula as in the Lie algebra case:

X(eH)=∑w∈Wε(w)ew(λ+ρ)(H)∑w∈Wε(w)ew(ρ)(H)displaystyle mathrm X (e^H)=frac sum _win Wvarepsilon (w)e^w(lambda +rho )(H)sum _win Wvarepsilon (w)e^w(rho )(H).

It should be noted, however, that Weyl's proof of the character formula in the compact group setting is completely different from the algebraic proof of the character formula in the setting of semisimple Lie algebras.^[7] Note also that in the compact group setting, it is common to use "real roots" and "real weights", which differ by a factor of idisplaystyle i from the roots and weights used here. Thus, the formula in the compact group setting has factors of idisplaystyle i in the exponent throughout.

The SU(2) case[edit]

In the case of the group SU(2), consider the irreducible representation of dimension m+1displaystyle m+1. If we take Tdisplaystyle T to be the diagonal subgroup of SU(2), the character formula in this case reads^[8]

X((eiθ00e−iθ))=ei(m+1)θ−e−i(m+1)θeiθ−e−iθ=sin((m+1)θ)sin(θ)displaystyle mathrm X left(beginpmatrixe^itheta &0\0&e^-itheta endpmatrixright)=frac e^i(m+1)theta -e^-i(m+1)theta e^itheta -e^-itheta =frac mathrm sin ((m+1)theta )mathrm sin (theta ).

(Both numerator and denominator in the character formula have two terms.) It is instructive to verify this formula directly in this case, so that we can observe the cancellation phenomenon implicit in the Weyl character formula.

Since the representations are known very explicitly, the character of the representation can be written down as

X((eiθ00e−iθ))=eimθ+ei(m−2)θ+⋯+e−imθdisplaystyle mathrm X left(beginpmatrixe^itheta &0\0&e^-itheta endpmatrixright)=e^imtheta +e^i(m-2)theta +cdots +e^-imtheta .

The Weyl denominator, meanwhile, is simply the function eiθ−e−iθdisplaystyle e^itheta -e^-itheta . Multiplying the character by the Weyl denominator gives

X((eiθ00e−iθ))(eiθ−e−iθ)=(ei(m+1)θ+ei(m−1)θ+⋯+e−i(m−1)θ)−(ei(m−1)θ+⋯+e−i(m−1)θ+e−i(m+1)θ)displaystyle mathrm X left(beginpmatrixe^itheta &0\0&e^-itheta endpmatrixright)(e^itheta -e^-itheta )=left(e^i(m+1)theta +e^i(m-1)theta +cdots +e^-i(m-1)theta right)-left(e^i(m-1)theta +cdots +e^-i(m-1)theta +e^-i(m+1)theta right).

We can now easily verify that most of the terms cancel between the two term on the right-hand side above, leaving us with only

X((eiθ00e−iθ))(eiθ−e−iθ)=ei(m+1)θ−e−i(m+1)θdisplaystyle mathrm X left(beginpmatrixe^itheta &0\0&e^-itheta endpmatrixright)(e^itheta -e^-itheta )=e^i(m+1)theta -e^-i(m+1)theta

so that

X((eiθ00e−iθ))=ei(m+1)θ−e−i(m+1)θeiθ−e−iθ=sin((m+1)θ)sin(θ)displaystyle mathrm X left(beginpmatrixe^itheta &0\0&e^-itheta endpmatrixright)=frac e^i(m+1)theta -e^-i(m+1)theta e^itheta -e^-itheta =frac mathrm sin ((m+1)theta )mathrm sin (theta ).

Note that the character in this case is a geometric series with R=e2iθdisplaystyle R=e^2itheta and that preceding argument is a small variant of the standard derivation of the formula for the sum of a finite geometric series.

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:^[9]

∑w∈Wε(w)ew(ρ)(H)=∏α∈Δ+(eα(H)/2−e−α(H)/2).displaystyle sum _win Wvarepsilon (w)e^w(rho )(H)=prod _alpha in Delta ^+(e^alpha (H)/2-e^-alpha (H)/2).

For special unitary groups, this is equivalent to the expression

∑σ∈Snsgn⁡(σ)X1σ(1)−1⋯Xnσ(n)−1=∏1≤i<j≤n(Xj−Xi)displaystyle sum _sigma in S_noperatorname sgn (sigma ),X_1^sigma (1)-1cdots X_n^sigma (n)-1=prod _1leq i<jleq n(X_j-X_i)

for the Vandermonde determinant.^[10]

Weyl dimension formula[edit]

By evaluating the character at H=0displaystyle H=0, Weyl's character formula gives the Weyl dimension formula

dim⁡(Vλ)=∏α∈Δ+(λ+ρ,α)∏α∈Δ+(ρ,α)displaystyle 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 Vλdisplaystyle V_lambda with highest weight λdisplaystyle lambda . (As usual, ρ is half the sum of the positive roots 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, using a version of L'Hospital's rule.^[11] In the SU(2) case described above, for example, we can recover the dimension m+1displaystyle m+1 of the representation by using L'Hospital's rule to evaluate the limit as θdisplaystyle theta tends to zero of sin((m+1)θ)/sin(θ)displaystyle mathrm sin ((m+1)theta )/mathrm sin (theta ).

We may consider as an example the complex semisimple Lie algebra sl(3,C), or equivalently the compact group SU(3). In that case, the representations are labeled by a pair (m1,m2)displaystyle (m_1,m_2) of non-negative integers. In this case, there are three positive roots and it is not hard to verify that the dimension formula takes the explicit form^[12]

dim(Vm1,m2)=12(m1+1)(m2+1)(m1+m2+2)displaystyle mathrm dim (V_m_1,m_2)=frac 12(m_1+1)(m_2+1)(m_1+m_2+2)

The case m1=1,m2=0displaystyle m_1=1,,m_2=0 is the standard representation and indeed the dimension formula gives the value 3 in this case.

Kostant multiplicity formula[edit]

The Weyl character formula gives the character of each representation as a quotient, where the numerator and denominator are each a finite linear combination of exponentials. While this formula in principle determines the character, it is not especially obvious how one can compute this quotient explicitly as a finite sum of exponentials. Already In the SU(2) case described above, it is not immediately obvious how to go from the Weyl character formula, which gives the character as sin((m+1)θ)/sin(θ)displaystyle mathrm sin ((m+1)theta )/mathrm sin (theta ) back to the formula for the character as a sum of exponentials:

eimθ+ei(m−2)θ+⋯+e−imθdisplaystyle e^imtheta +e^i(m-2)theta +cdots +e^-imtheta .

In this case, it is perhaps not terribly difficult to recognize the expression sin((m+1)θ)/sin(θ)displaystyle mathrm sin ((m+1)theta )/mathrm sin (theta ) as the sum of a finite geometric series, but in general we need a more systematic procedure.

In general, the division process can be accomplished by computing a formal reciprocal of the Weyl denominator and then multiplying the numerator in the Weyl character formula by this formal reciprocal.^[13] The result gives the character as a finite sum of exponentials. The coefficients of this expansion are the dimensions of the weight spaces, that is, the multiplicities of the weights. We thus obtain from the Weyl character formula a formula for the multiplicities of the weights, known as the Kostant multiplicity formula. An alternative formula, that is more computationally tractable in some cases, is given in the next section.

Freudenthal's formula[edit]

Hans Freudenthal's formula is a recursive formula for the weight multiplicities that gives the same answer as the Kostant multiplicity formula, but is sometimes
easier to use for calculations as there can be far fewer terms to sum. The formula is based on use of the Casimir element and its derivation is independent of the character formula.
It states^[14]

(‖Λ+ρ‖2−‖λ+ρ‖2)mΛ(λ)=2∑α∈Δ+∑j≥1(λ+jα,α)mΛ(λ+jα)displaystyle (

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[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

∏m=1∞(1−x2m)(1−x2m−1y)(1−x2m−1y−1)=∑n=−∞∞(−1)nxn2yn.displaystyle prod _m=1^infty left(1-x^2mright)left(1-x^2m-1yright)left(1-x^2m-1y^-1right)=sum _n=-infty ^infty (-1)^nx^n^2y^n.

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

∑w∈W(−1)ℓ(w)w(eλ+ρS)eρ∏α∈Δ+(1−e−α).displaystyle sum _win 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=∑I(−1)|I|eΣIdisplaystyle S=sum _I(-1)^Ie^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)=(1p−1q)∏n,m=1∞(1−pnqm)cnmdisplaystyle j(p)-j(q)=left(1 over p-1 over qright)prod _n,m=1^infty (1-p^nq^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:

(β,β−2ρ)cβ=∑γ+δ=β(γ,δ)cγcδdisplaystyle (beta ,beta -2rho )c_beta =sum _gamma +delta =beta (gamma ,delta )c_gamma c_delta ,

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

cβ=∑n≥1mult⁡(β/n)n.displaystyle c_beta =sum _ngeq 1operatorname 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 πdisplaystyle pi is an irreducible, admissible representation of a real, reductive group G with infinitesimal character λdisplaystyle lambda . Let Θπdisplaystyle Theta _pi be the Harish-Chandra character of πdisplaystyle 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

Θπ|H′=∑w∈W/Wλawewλeρ∏α∈Δ+(1−e−α)._H'=sum _win W/W_lambda a_we^wlambda over e^rho prod _alpha in Delta ^+(1-e^-alpha ).

Here

• W is the complex Weyl group of HCdisplaystyle H_mathbb C with respect to GCdisplaystyle G_mathbb C
  


• 
  Wλdisplaystyle W_lambda is the stabilizer of λdisplaystyle lambda in W

and the rest of the notation is as above.

The coefficients awdisplaystyle 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]

• Character theory


• Algebraic character


• Demazure character formula

References[edit]

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  Hall, Brian C. (2015), Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 978-3319134666.mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
  


• 
  Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90053-7.


• 
  Infinite dimensional Lie algebras, V. G. Kac,
  ISBN 0-521-37215-1
  


• 
  Duncan J. Melville (2001) [1994], "Weyl–Kac character formula", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, 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