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.
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.
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.
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.
be an irreducible, finite-dimensional representation of a complex semisimple Lie algebra
By elementary considerations, the character may be computed as where the sum ranges over all the weights
may also be computed as where Using the Weyl denominator formula (described below), the character formula may be rewritten as or, equivalently, The character is itself a large 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.
and the highest weight from the Weyl denominator) and things in the Weyl-group orbit of
and the Peter–Weyl theorem asserts that the characters form an orthonormal basis for the space of square-integrable class functions on
is then given by the same formula as in the Lie algebra case: 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.
Thus, the formula in the compact group setting has factors of
In the case of the group SU(2), consider the irreducible representation of dimension
to be the diagonal subgroup of SU(2), the character formula in this case reads[8] (Both numerator and denominator in the character formula have two terms.)
Multiplying the character by the Weyl denominator gives 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 so that The character in this case is a geometric series with
and that preceding argument is a small variant of the standard derivation of the formula for the sum of a finite geometric series.
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] For special unitary groups, this is equivalent to the expression for the Vandermonde determinant.
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'Hôpital's rule.
We may consider as an example the complex semisimple Lie algebra sl(3,C), or equivalently the compact group SU(3).
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.
back to the formula for the character as a sum of exponentials: In this case, it is perhaps not terribly difficult to recognize the expression
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.
An alternative formula, that is more computationally tractable in some cases, is given in the next section.
In the simplest case of the affine Lie algebra of type A1 this is the Jacobi triple product identity The character formula can also be extended to integrable highest weight representations of generalized Kac–Moody algebras, when the character is given by Here S is a correction term given in terms of the imaginary simple roots by 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 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: where the sum is over positive roots γ, δ, and Harish-Chandra showed that Weyl's character formula admits a generalization to representations of a real, reductive group.
is an irreducible, admissible representation of a real, reductive group G with infinitesimal character
If H is a Cartan subgroup of G and H' is the set of regular elements in H, then Here and the rest of the notation is as above.
Results on these coefficients may be found in papers of Herb, Adams, Schmid, and Schmid-Vilonen among others.