Application of the Weyl Character Formula and Examples of Calculation
Introduction
The Weyl Character Formula is a cornerstone of modern Lie theory, providing a precise method to compute the characters of irreducible representations of complex semisimple Lie groups. This formula connects the symmetric functions of roots to the structure of the group, enabling the classification of representations through their characters. Its derivation relies on the theory of symmetric polynomials and the properties of root systems, making it a powerful tool in both theoretical and applied mathematics. This article explores the formula’s derivation, its application to specific Lie groups, and illustrative examples of its computation.
Core Content
Derivation of the Weyl Character Formula
The Weyl Character Formula arises from the study of the symmetric group and the representation theory of Lie groups. For a complex semisimple Lie group $ G $ with root system $ \Phi $, the formula expresses the character $ \chi(\lambda) $ of an irreducible representation with highest weight $ \lambda $ as:
$$
\chi(\lambda) = \frac{1}{|W|} \sum_{w \in W} \mathrm{e}^{2\pi i \langle \lambda, w \rangle}
$$
where $ W $ is the Weyl group associated with $ \Phi $, and $ \langle \lambda, w \rangle $ denotes the inner product of $ \lambda $ with the Weyl group element $ w $. The formula simplifies when $ \lambda $ is a dominant weight, as the sum over $ W $ reduces to a symmetric function of the roots.
The derivation involves two key steps:
- Symmetric Polynomial Expansion: The character is expressed as a symmetric function of the roots, leveraging the fact that the group action is transitive.
- Weyl Group Symmetry: The formula accounts for the symmetries of the root system, ensuring that the character is invariant under conjugation by elements of $ W $.
Application to the Symmetric Group
For the symmetric group $ S_n $, the Weyl Character Formula simplifies significantly. The group has a root system corresponding to the symmetric group, and the formula reduces to a sum over permutations. For example, the character of the representation corresponding to the permutation module is given by:
$$
\chi(\lambda) = \frac{1}{n!} \sum_{\sigma \in S_n} \mathrm{e}^{2\pi i \langle \lambda, \sigma \rangle}
$$
This formula is particularly useful in computing the dimensions of irreducible representations, as the character values directly relate to the representation’s dimension.
Application to Lie Groups
The Weyl Character Formula is extended to Lie groups such as $ \mathrm{SU}(2) $, $ \mathrm{SU}(3) $, and $ \mathrm{SL}(n) $. For $ \mathrm{SU}(2) $, the formula reduces to a simple expression involving the angular momentum eigenvalues. For example, the character of the irreducible representation with highest weight $ \lambda $ is:
$$
\chi(\lambda) = \frac{1}{2} \left( \frac{1}{\sqrt{1 - \lambda^2}} \right)
$$
This result is derived from the root system of $ \mathrm{SU}(2) $, where the roots are $ \pm \lambda $, and the Weyl group acts as the symmetric group $ S_2 $.
Similarly, for $ \mathrm{SU}(3) $, the formula involves more complex symmetric polynomials and requires the use of the Weyl group’s action on the roots. The character of the representation corresponding to the highest weight $ \lambda = (2,1) $ is computed as:
$$
\chi(\lambda) = \frac{1}{3} \left( \mathrm{e}^{2\pi i \langle \lambda, \sigma \rangle} + 2 \right)
$$
where $ \sigma $ are the elements of the Weyl group $ S_3 $. This example illustrates how the formula adapts to higher-dimensional Lie groups.
Computational Examples
To demonstrate the formula’s utility, consider the case of $ \mathrm{SL}(2) $, where the root system is $ \Phi = { \pm \alpha } $. The character of the irreducible representation with highest weight $ \lambda = \alpha $ is:
$$
\chi(\lambda) = \frac{1}{2} \left( 1 + \mathrm{e}^{2\pi i \langle \lambda, \sigma \rangle} \right)
$$
For $ \sigma \in W $, the inner product simplifies to $ \langle \lambda, \sigma \rangle = \pm 1 $, leading to:
$$
\chi(\lambda) = \frac{1}{2} (1 + \mathrm{e}^{2\pi i \pm 1}) = \frac{1}{2} (1 \pm 1)
$$
This results in $ \chi(\lambda) = 0 $ or $ 1 $, depending on the sign, reflecting the representation’s structure.
Another example involves $ \mathrm{SL}(3) $, where the formula requires computing the symmetric sum over the Weyl group. For the highest weight $ \lambda = (2,1) $, the character is:
$$
\chi(\lambda) = \frac{1}{6} \left( \mathrm{e}^{2\pi i \langle \lambda, \sigma_1 \rangle} + \mathrm{e}^{2\pi i \langle \lambda, \sigma_2 \rangle} + \mathrm{e}^{2\pi i \langle \lambda, \sigma_3 \rangle} \right)
$$
where $ \sigma_1, \sigma_2, \sigma_3 $ are the simple reflections. This computation highlights the complexity of applying the formula to higher-dimensional groups.
Applications in Representation Theory
The Weyl Character Formula is pivotal in classifying irreducible representations of Lie groups. By computing characters, one can determine the dimensions of representations and their decomposition into irreducible components. For instance, in the case of $ \mathrm{SU}(3) $, the formula allows the determination of the multiplicities of irreducible representations through the use of the Weyl character ring.
Moreover, the formula plays a critical role in the Langlands program, where it is used to relate automorphic forms to representations of Lie groups. Its application in physics, particularly in quantum mechanics, aids in the study of symmetry groups and the classification of particles.
Conclusion
The Weyl Character Formula provides a systematic approach to computing the characters of irreducible representations of complex semisimple Lie groups. Its derivation relies on symmetric polynomial theory and the properties of root systems, making it a foundational tool in Lie theory. By applying the formula to specific groups like $ \mathrm{SU}(