Fulton And Harris: Exercise 1.3 In Section 1.1

Introduction to Representation Theory

In the realm of abstract algebra, representation theory plays a vital role in understanding the properties and behavior of finite groups. The concept of representation theory was first introduced by the mathematician William Burnside in the early 20th century. It has since become a fundamental tool in the study of group theory, with applications in various fields such as physics, computer science, and engineering.

Finite Groups and Vector Spaces

In this exercise, we are given a finite group $G$ and an $n$-dimensional $\mathbb C$-vector space $V$. We are also given a group homomorphism $\rho: G \to \text{GL}(V)$, where $\text{GL}(V)$ is the general linear group of $V$. The group homomorphism $\rho$ is a representation of $G$ on $V$, and it is a fundamental concept in representation theory.

Exercise 1.3: Representation of a Finite Group

Let $G$ be a finite group, let $V$ be an $n$-dimensional $\mathbb C$-vector space, and let $\rho: G \to \text{GL}(V)$ be a group homomorphism. We are asked to show that the representation $\rho$ is equivalent to a representation of $G$ on a vector space of the same dimension.

Step 1: Define the Character of the Representation

The character of the representation $\rho$ is defined as the function $\chi: G \to \mathbb C$ given by $\chi(g) = \text{tr}(\rho(g))$, where $\text{tr}(\rho(g))$ is the trace of the matrix $\rho(g)$. The character of the representation is a fundamental concept in representation theory, and it plays a crucial role in the study of group representations.

Step 2: Show that the Character is a Class Function

We need to show that the character $\chi$ is a class function, meaning that it is constant on conjugacy classes of $G$. Let $g, h \in G$ be two elements that are conjugate, i.e., there exists $x \in G$ such that $g = xhx^{-1}$. We need to show that $\chi(g) = \chi(h)$.

Step 3: Use the Group Homomorphism Property

Since $\rho$ is a group homomorphism, we have $\rho(g) = \rho(xhx^{-1}) = \rho(x)\rho(h)\rho(x)^{-1}$. Taking the trace of both sides, we get $\text{tr}(\rho(g)) = \text{tr}(\rho(x)\rho(h)\rho(x)^{-1})$. Using the property of the trace, we can rewrite this as $\text{tr}(\rho(g)) = \text{tr}(\rho(h))$.

Step 4: Conclude that the Character is a Class Function

Since $\text{tr}(\rho(g)) = \text{tr}(\rho(h))$, we have $\chi(g) = \chi(h)$. This shows that the character $\chi$ is a class function, that it is constant on conjugacy classes of $G$.

Step 5: Show that the Representation is Equivalent to a Representation of the Same Dimension

We need to show that the representation $\rho$ is equivalent to a representation of $G$ on a vector space of the same dimension. Let $W$ be a vector space of dimension $n$ and let $\sigma: G \to \text{GL}(W)$ be a representation of $G$ on $W$. We need to show that there exists an isomorphism $\phi: V \to W$ such that $\sigma(g) = \phi \rho(g) \phi^{-1}$ for all $g \in G$.

Step 6: Use the Character to Show Equivalence

Since the character $\chi$ is a class function, we have $\chi(g) = \chi(h)$ for all $g, h \in G$ that are conjugate. This means that the character $\chi$ is constant on conjugacy classes of $G$. Let $\chi_0$ be the value of the character $\chi$ on the identity element of $G$. We can define a new representation $\sigma: G \to \text{GL}(W)$ by $\sigma(g) = \chi_0^{-1} \rho(g)$. This representation has the same character as the original representation $\rho$, and it is equivalent to the original representation.

Conclusion

In this exercise, we have shown that the representation $\rho$ is equivalent to a representation of $G$ on a vector space of the same dimension. We have used the character of the representation to show that the representation is equivalent to a representation of the same dimension. This result is a fundamental concept in representation theory, and it has important implications for the study of group representations.

References

• Fulton, W., & Harris, J. (1991). Representation theory: A first course. Springer-Verlag.
• Serre, J.-P. (1977). Linear representations of finite groups. Springer-Verlag.

Further Reading

• Artin, E. (1947). Galois theory. Notre Dame Mathematical Lectures, 2.
• Lang, S. (1965). Algebra. Addison-Wesley.
• Serre, J.-P. (1977). Linear representations of finite groups. Springer-Verlag.
  Fulton and Harris: Exercise 1.3 in Section 1.1 - Q&A =====================================================

Introduction to Representation Theory

In the previous article, we discussed Exercise 1.3 in Section 1.1 of Fulton and Harris' book on Representation Theory. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q: What is the significance of the character of a representation?

A: The character of a representation is a fundamental concept in representation theory. It is a function that assigns a complex number to each element of the group, and it encodes important information about the representation. The character is used to classify representations and to study their properties.

Q: Why is the character a class function?

A: The character is a class function because it is constant on conjugacy classes of the group. This means that if two elements of the group are conjugate, then their characters are equal. This property is crucial in the study of representation theory.

Q: How do we show that the representation is equivalent to a representation of the same dimension?

A: We show that the representation is equivalent to a representation of the same dimension by using the character of the representation. We define a new representation with the same character as the original representation, and we show that it is equivalent to the original representation.

Q: What is the importance of the result in Exercise 1.3?

A: The result in Exercise 1.3 is important because it shows that any representation of a finite group can be reduced to a representation of the same dimension. This result has far-reaching implications for the study of representation theory and its applications.

Q: How does Exercise 1.3 relate to other areas of mathematics?

A: Exercise 1.3 is related to other areas of mathematics, such as algebraic geometry and number theory. The concepts and techniques developed in Exercise 1.3 have applications in these areas and provide a deeper understanding of the underlying mathematics.

Q: What are some common mistakes to avoid when working with representations?

A: Some common mistakes to avoid when working with representations include:

• Not checking that the representation is well-defined
• Not verifying that the representation is a homomorphism
• Not using the correct properties of the representation
• Not being careful with the indexing of the representation

Q: How can I apply the concepts and techniques developed in Exercise 1.3 to real-world problems?

A: The concepts and techniques developed in Exercise 1.3 can be applied to real-world problems in various fields, such as physics, engineering, and computer science. For example, representation theory is used in the study of symmetry in physics and in the design of algorithms for computer science.

Conclusion

In this Q&A article, we have addressed some common questions and doubts that readers may have about Exercise 1.3 in Section 1.1 of Fulton and Harris' book on Representation Theory. We hope that this article has provided a deeper understanding of the concepts and techniques developed in Exercise 1.3 and has inspired readers to explore the fascinating world of representation theory.

References

• Fulton, W., & Harris, J. (1991). Representation theory: A first course. Springer-Verlag.
• Serre, J.-P. (1977). Linear representations of finite groups. Springer-Verlag.
• Artin, E. (1947). Galois theory. Notre Dame Mathematical Lectures, 2.
• Lang, S. (1965). Algebra. Addison-Wesley.

Further Reading

• Serre, J.-P. (1977). Linear representations of finite groups. Springer-Verlag.
• Fulton, W., & Harris, J. (1991). Representation theory: A first course. Springer-Verlag.
• Humphreys, J. E. (1972). Introduction to Lie algebras and representation theory. Springer-Verlag.