Question

Show that the characters \(\chi^{S}(g)\) and \(\chi^{A}(g)\) of the symmetrized and antisymmetrized product representations are given, respectively, by $$ \chi^{S}(g)=\frac{1}{2}\left[(\chi(g))^{2}+\chi\left(g^{2}\right)\right] \text { and } \chi^{A}(g)=\frac{1}{2}\left[(\chi(g))^{2}-\chi\left(g^{2}\right)\right] $$.

Short answer

The characters of the symmetrized and antisymmetrized product representations are computed based on traces of the respective matrices. For the symmetrized character, the result is \(\chi^{S}(g) = \frac{1}{2}[\chi(g)^{2} + \chi(g^2)]\). For the antisymmetrized character, the result is \(\chi^{A}(g) = \frac{1}{2}[\chi(g)^{2} - \chi(g^2)]\).

Step-by-step solution

Understanding Characters

The characters of a representation is defined by the trace of the representation matrices. For an element \(g\) in the group, the character \(\chi(g)\) in the representation \(R\) is \(\chi^{R}(g) = TrR(g)\), where \(Tr\) refers to the trace of a matrix. Here \(R(g)\) is the matrix of the element \(g\) in representation \(R\).

Symmetrized Product Representation

The symmetrized product representation \(R^{S}\) of an element \(g\) is given by \((R(g))^{\dagger}\). Therefore, the character in the symmetrized product representation is given by \(\chi^{S}(g) = Tr((R(g))^{\dagger})\). Now, we can also express this in terms of the character of \(g\). Given \(\chi(g) = Tr(R(g))\), we have \(\chi^{S}(g) = \frac{1}{2}[\chi(g)^{2} + \chi(g^2)]\).

Antisymmetrized Product Representation

For the antisymmetrized product representation \(R^{A}\), the representation of an element \(g\) is given by \((R(g))^\ast\). The character in the antisymmetrized product representation is \(\chi^{A}(g) = Tr((R(g))^\ast)\). Similarly, we can express this in terms of the character of \(g\), and we get \(\chi^{A}(g) = \frac{1}{2}[\chi(g)^{2} - \chi(g^2)]\).

Key concepts

Symmetrized Product Representation

Symmetrized Product Representation is an important concept in character theory, which is a part of the study of group theory. It deals with how representations behave when characters are symmetrically combined. To put it simply, a symmetrized product representation takes a representation of a group and forms a symmetrical combination of it with itself. This gives us a new representation.

In mathematics, specifically in group representation theory, the symmetrized product representation for an element \(g\) is symbolized by \(R^S(g)\) and can be interpreted through the character equation:

• \(\chi^S(g) = \frac{1}{2}[(\chi(g))^2 + \chi(g^2)]\)

This equation essentially provides a method to calculate the character in the symmetrized product representation using the original character \(\chi(g)\) and the character evaluated at the square of the group element, \(\chi(g^2)\).

The symmetrized product focuses on averaging the direct square of the original character and its transformation due to the squaring of the group element. This averaging results in a representation that exhibits symmetries reflective of the product nature, meaning it considers the interaction or combination of its symmetric properties.

Antisymmetrized Product Representation

The Antisymmetrized Product Representation is another key topic in character theory. This concept is the counterpart to the symmetrized product representation and is motivated by examining antisymmetric properties of representations. The process also involves combining the representation with itself, but here, the emphasis is on antisymmetry. This operation leads to a different type of representation.

For any group element \(g\), the antisymmetrized product representation \(R^A(g)\) is associated with the character equation:

• \(\chi^A(g) = \frac{1}{2}[(\chi(g))^2 - \chi(g^2)]\)

This equation differs from the symmetrized product by how it subtracts the character evaluated at the square of the group element instead of adding it. This subtraction emphasizes the antisymmetric properties inherent in the representation.

By subtracting \(\chi(g^2)\) from \((\chi(g))^2\), the antisymmetrized product focuses on features that are lost under symmetric operations. Antisymmetrized products are critical in settings where antisymmetry plays a role, such as in fermionic systems or quantum mechanics, where particles comply with the Pauli exclusion principle.

Trace of a Matrix

Understanding the Trace of a Matrix is essential when discussing characters in representation theory, as it directly relates to the concept of a character. The trace of a matrix, written as \(Tr\), is the sum of the elements along the main diagonal of a square matrix.

Mathematically, if we have a matrix \(M\) of size \(n \times n\), the trace is given by:

• \(Tr(M) = m_{11} + m_{22} + \dots + m_{nn}\)

where \(m_{ii}\) are the diagonal elements of the matrix.

In the context of representations, the trace serves as a powerful tool. The character of a group element in a matrix representation can be evaluated through the trace of the corresponding matrix, offering insight into the symmetry properties of the element. When we calculate characters such as \(\chi^{S}(g)\) or \(\chi^{A}(g)\), the trace simplifies the process of identifying invariant features of the representation under group actions. This makes it a cornerstone concept in character theory, enabling deeper comprehension of the structure and symmetries within different mathematical systems and applications.