χ

Definition

In representation theory, the symbol χ represents the character of a representation of a group. It is a function that assigns to each group element a complex number, which reflects the trace of the corresponding group element in the representation matrix. Characters help in understanding the structure of representations and are central to the study of how groups act on vector spaces.

5 Must Know Facts For Your Next Test

1. Characters are constant on conjugacy classes, meaning that if two group elements are conjugate, they have the same character value.
2. The inner product of two characters can be computed using their values at different group elements, and this inner product is used to determine orthogonality between characters.
3. The sum of the squares of the dimensions of all irreducible representations of a group equals the order of the group, which can be derived from properties of characters.
4. Characters provide powerful tools for classifying representations and determining whether they are equivalent or not.
5. For finite groups, characters form an orthonormal basis for functions on the group when viewed as complex-valued functions.

Review Questions

• How do characters facilitate understanding the structure of representations in group theory?
  • Characters provide a way to encapsulate important information about representations through a single function for each group element. This allows mathematicians to analyze and classify representations without directly dealing with matrices or complicated computations. The character's properties, like being constant on conjugacy classes and forming an orthonormal basis, make them essential tools in exploring the equivalence and irreducibility of representations.
• Discuss how orthogonality relations for characters impact the study of representation theory.
  • Orthogonality relations indicate that different irreducible characters are orthogonal to each other under an inner product defined on character values. This implies that one can distinguish between different representations by their characters. Consequently, these relations help identify how many distinct irreducible representations exist for a given group and provide a method to calculate multiplicities when decomposing representations into irreducibles.
• Evaluate the implications of characters being constant on conjugacy classes for representation theory and its applications.
  • The fact that characters are constant on conjugacy classes simplifies many calculations in representation theory, allowing researchers to work with fewer elements. This property means that instead of analyzing every single element of a group, one can focus on representatives from each conjugacy class. The implications extend to applications in areas such as physics and chemistry, where understanding symmetries and transformations is essential; characters help predict behavior under group actions by linking algebraic properties with geometric interpretations.

Related terms

Trace: The trace of a matrix is the sum of its diagonal elements, providing important information about the matrix's properties and its associated linear transformation.

Irreducible Representation:

An irreducible representation is one that has no nontrivial invariant subspaces, meaning it cannot be decomposed into smaller representations.

Orthogonality Relations:

Orthogonality relations are mathematical statements about the inner products of characters of representations, which establish important relationships between different characters in a group.