# Irreducible Representations and Class

• sineontheline
In summary, irreducible representations are mathematical objects used to describe the symmetry properties of a physical system, and they play a crucial role in understanding the behavior of quantum mechanical systems. They provide a convenient way to classify and describe symmetries and are closely related to group theory. The difference between reducible and irreducible representations is that reducible ones can be broken down, while irreducible ones cannot. Irreducible representations are also used to determine the character table of a group, which summarizes the properties of the group's irreducible representations and helps analyze the symmetry of a physical system.
sineontheline
So I'm studying molecules and symmetry and I was wondering if there was a intuitive way of understanding why there are as many irreducible representations as there are classes. I keep getting lost in the math of the characters.

*bump* :)

## 1. What are irreducible representations?

Irreducible representations are mathematical objects used to describe the symmetry properties of a physical system. They are commonly used in the field of group theory and play a crucial role in understanding the behavior of quantum mechanical systems.

## 2. What is the significance of irreducible representations in physics?

Irreducible representations provide a convenient way to classify and describe the symmetries of a physical system. They are essential in understanding the energy levels, selection rules, and other properties of molecules, crystals, and other quantum systems.

## 3. How are irreducible representations related to group theory?

Irreducible representations are closely related to group theory, which is the mathematical study of symmetry. In group theory, a group is a set of elements that can be combined (or operated) in a certain way to produce other elements of the group. Irreducible representations are used to describe how these operations transform the elements of the group.

## 4. What is the difference between reducible and irreducible representations?

Reducible representations can be broken down into a combination of two or more irreducible representations, while irreducible representations cannot be further simplified. In other words, reducible representations contain multiple independent subrepresentations, while irreducible representations are self-contained and cannot be decomposed into smaller parts.

## 5. How are irreducible representations used to determine the character table of a group?

The character table of a group is a square matrix that summarizes the properties of the group's irreducible representations. Each row and column in the character table corresponds to a particular irreducible representation, and the values in the table represent how the group's operations transform the basis elements of each representation. By finding the irreducible representations and their corresponding characters, the character table can be constructed and used to analyze the symmetry of a physical system.

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