The Weyl character formula, the half-spin representations, and equal rank subgroups

Area of Science:

• Lie algebra representation theory
• Harmonic analysis
• Mathematical physics

Background:

• The study of Lie algebras and their representations is fundamental in mathematics and physics.
• The Weyl character formula is a cornerstone in understanding irreducible representations of semi-simple Lie algebras.
• Understanding the relationship between representations of a Lie algebra and its subalgebras is crucial for various applications.

Purpose of the Study:

• To generalize the Weyl character formula for reductive Lie subalgebras.
• To establish a connection between the irreducible representations of a semi-simple Lie algebra and its reductive subalgebra of the same rank.
• To provide a new formula for computing characters of irreducible representations.

Main Methods:

• Assigning a multiplet of irreducible representations of the subalgebra B to each irreducible representation of the larger Lie algebra F.
• Utilizing the index of the Weyl group of B in the Weyl group of F to determine the size of these multiplets.
• Developing a generalized formula based on alternating sums of characters.

Main Results:

• A novel generalization of the Weyl character formula is derived.
• The formula expresses the character of an irreducible representation of F as a quotient of alternating sums of characters from associated multiplets.
• The numerator involves characters of the multiplet associated with the representation Vlambda, and the denominator involves characters of the multiplet associated with the trivial representation.

Conclusions:

• The generalized formula provides a powerful tool for studying representations of reductive Lie subalgebras.
• This work deepens the understanding of the structure of Lie algebra representations and their relationships.
• The findings have potential implications in areas such as quantum mechanics and particle physics where Lie algebras are extensively used.