Brauer characters

Definition

Brauer characters are a generalization of the concept of characters in representation theory, specifically designed for non-associative algebras. They play a crucial role in understanding the representations of finite groups and their associated algebras over fields, particularly when dealing with modular representations. These characters provide insight into the structure and properties of the algebra, allowing for a deeper analysis of its representations and their dimensions.

5 Must Know Facts For Your Next Test

1. Brauer characters arise when studying representations of finite groups over fields with positive characteristic, particularly in situations where traditional character theory doesn't apply.
2. They can be constructed from projective modules in non-associative algebras and provide important information about their irreducible representations.
3. Brauer characters are closely related to the Brauer group, which classifies the equivalence classes of central simple algebras over a field.
4. The values of Brauer characters are often linked to the dimensions of irreducible representations, helping to determine how these representations behave under various operations.
5. In the context of modular representation theory, Brauer characters help solve problems related to the decomposition of representations into irreducible components.

Review Questions

• How do Brauer characters extend traditional character theory in the context of non-associative algebras?
  • Brauer characters extend traditional character theory by providing a framework to analyze representations of non-associative algebras, especially over fields with positive characteristic. While classical character theory is effective for associative algebras, it falls short in modular contexts. Brauer characters fill this gap by incorporating projective modules and allowing for a richer understanding of representation dimensions and their structures within these algebras.
• Discuss the significance of Brauer characters in understanding modular representations and their irreducible components.
  • Brauer characters are significant in modular representations as they help illuminate the structure of these representations when the field's characteristic divides the group order. They provide essential insights into how these representations can be decomposed into irreducible components. By analyzing Brauer characters, mathematicians can better understand the relationships between different modules and the behavior of these representations under various algebraic operations.
• Evaluate how Brauer characters can be applied to solve problems related to central simple algebras and their classifications.
  • Brauer characters can be applied to classify central simple algebras by connecting them with equivalence classes through the Brauer group. By studying these characters, one can gain insights into the structure and relationships among central simple algebras over a given field. This connection allows for a deeper understanding of their representation theory, leading to potential solutions for problems involving module decompositions and character values, which are crucial for further advancements in both algebra and number theory.

Related terms

Representation theory: The study of how algebraic structures, like groups or algebras, can be represented through linear transformations on vector spaces.

Modular representation: A representation of a group over a field whose characteristic divides the order of the group, which often leads to unique properties and challenges.

Character theory: A branch of mathematics that studies the representation of groups through characters, which are functions that assign complex numbers to group elements in a way that preserves group structure.