# Constructing Character Tables: Methods & Invariant Subgroups

• CPL.Luke
In summary, constructing the character table for a group involves finding irreducible representations and utilizing them to determine representations for more complicated groups. This process can be found in the book "Group Theory" by Hammermesh. There are also alternative methods available. If the group contains an invariant subgroup, this may affect the character table.
CPL.Luke
how does one go about constructing the character table for a group? while only knowing the group, and the classes. I was thinking of starting to construct representations and diagonalizing them but that seems like it would take an exorbitant amount of time.

is there a method for constructing the character table without knowing the irreducible representations?sorry if this should be moved, this is related to a homework problem, but as its talking more about methods I thought it could go here.Also on asimilar note if the group contains an invariant subgroup how does this affect the character table

Edit: I realized this is a physicist thing, but the character is the trace of a matrix representation, and the class is what mathematicians know as a conjugacy class

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One starts by finding irreducible representations for the simplest groups first, the Abelian groups, where the number of classes is equal to the order of the group. Then one works up to more complicated groups, utilizing previously determined
representations for the subgroups. This procedure is carried out systematically for many groups in the book "Group Theory" by Hammermesh (Addison-Wesley, 1962), Chapter 4. Several alternative methods are considered. I cannot elaborate more here but hope this might point you in a useful direction. Good luck!

## What is a character table?

A character table is a table that summarizes the properties and behaviors of the irreducible representations of a group. It contains information about the characters, dimensions, and symmetry properties of the representations, and can be used to analyze the group's structure and properties.

## Why is it important to construct a character table?

Constructing a character table allows us to understand the structure and properties of a group. It can also help us identify important subgroups and study their relationships. Character tables are useful tools in many areas of mathematics and physics, including group theory, quantum mechanics, and crystallography.

## What are the methods for constructing a character table?

There are several methods for constructing a character table, including the direct method, the projection method, and the reduction method. The direct method involves calculating the characters directly from the group's multiplication table. The projection method uses characters from a smaller group to construct the character table for a larger group. The reduction method involves breaking down a large group into smaller, easier-to-analyze subgroups.

## What is an invariant subgroup?

An invariant subgroup is a subgroup of a group that remains unchanged under certain operations. In other words, if an element of the group is applied to any element of the subgroup, the resulting element will still be in the subgroup. Invariant subgroups are important when constructing character tables because they can help simplify the analysis of a group's structure.

## How can character tables be used in practice?

Character tables are used in many areas of mathematics and physics, including crystallography, quantum mechanics, and molecular symmetry. They can help us understand the symmetry properties of molecules, predict the properties of materials, and analyze the behavior of quantum mechanical systems. In addition, character tables are useful for solving problems in number theory, geometry, and other areas of mathematics.

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