Equivalent Representations

Oct 13, 2025 by magicalceylontours.com 45 views

Introduction

In the realm of representation theory, two representations are considered equivalent if there exists an equivariant map between them. However, this definition can be quite abstract and difficult to work with. In this article, we will show that two representations are equivalent if and only if their matrices are equal in some bases. This result provides a more concrete and intuitive understanding of equivalence in representation theory.

Preliminaries

Before we dive into the main result, let's recall some basic definitions and concepts.

Representations

A representation of a group $G$ is a pair $(\pi, V)$ , where $V$ is a vector space and $\pi: G \to \text{GL}(V)$ is a homomorphism from $G$ to the general linear group of $V$ . In other words, $\pi$ is a way of assigning a linear transformation to each element of $G$ in a way that respects the group operation.

Equivariant Maps

An equivariant map between two representations $(\pi_1, V_1)$ and $(\pi_2, V_2)$ is a linear map $f: V_1 \to V_2$ such that $f(\pi_1(g)v) = \pi_2(g)f(v)$ for all $g \in G$ and $v \in V_1$ . In other words, the map $f$ commutes with the action of $G$ .

Equivalent Representations

Two representations $(\pi_1, V_1)$ and $(\pi_2, V_2)$ are equivalent if there exists an equivariant map $f: V_1 \to V_2$ that is both injective and surjective. In this case, we say that $f$ is an isomorphism between the two representations.

The Main Result

We are now ready to state the main result:

Theorem 1. Two representations $(\pi_1, V_1)$ and $(\pi_2, V_2)$ are equivalent if and only if there exist bases $B_1$ and $B_2$ such that $[\pi_1]_{B_1} = [\pi_2]_{B_2}$ .

Proof

To prove this result, we will first assume that the representations are equivalent and show that there exist bases $B_1$ and $B_2$ such that $[\pi_1]_{B_1} = [\pi_2]_{B_2}$ . Then, we will assume that there exist bases $B_1$ and $B_2$ such that $[\pi_1]_{B_1} = [\pi_2]_{B_2}$ and show that the representations are equivalent.

Step 1: Equivalence Implies Equal Matrices

Let $(\pi_1, V_1)$ and $(\pi_2, V_2)$ be equivalent representations. Then, there exists an equivariant map $f: V_1 \to V_2$ that is both injective and surjective. Let $B_1$ be a basis for $V_1$ and let $B_2$ be a basis for $V_2$ . We can extend $f$ to a linear map $F: \mathbb{R}^n \to \mathbb{R}^m$ by defining $F(e_i) = f(e_i)$ for each $i$ . Then, $F$ is an isomorphism between the two representations.

Now, let $A$ be the matrix of $f$ with respect to the bases $B_1$ and $B_2$ . Then, $A$ is a square matrix of size $n \times m$ . We can write $A = (a_{ij})$ , where $a_{ij}$ is the entry in the $i$ th row and $j$ th column of $A$ .

We claim that $A$ is invertible. To see this, let $v \in V_1$ be a vector and let $w = f(v) \in V_2$ be its image under $f$ . Then, we can write $v = \sum_{i=1}^n c_i e_i$ for some scalars $c_i$ . We have

f(v) = f\left(\sum_{i=1}^n c_i e_i\right) = \sum_{i=1}^n c_i f(e_i) = \sum_{i=1}^n c_i A e_i = A \left(\sum_{i=1}^n c_i e_i\right) = A v.

This shows that $A$ is invertible.

Now, let $B_1 = \{e_1, \ldots, e_n\}$ be a basis for $V_1$ and let $B_2 = \{f(e_1), \ldots, f(e_n)\}$ be a basis for $V_2$ . Then, we have

[\pi_1]_{B_1} = [\pi_2]_{B_2}.

This completes the first part of the proof.

Step 2: Equal Matrices Implies Equivalence

Let $(\pi_1, V_1)$ and $(\pi_2, V_2)$ be representations and let $B_1$ and $B_2$ be bases such that $[\pi_1]_{B_1} = [\pi_2]_{B_2}$ . We need to show that the representations are equivalent.

Let $A$ be the matrix of $\pi_1$ with respect to the basis $B_1$ . Then, we have

[\pi_1]_{B_1} = A.

Similarly, let $B$ be the matrix of $\pi_2$ with respect to the basis $B_2$ . Then, we have

[\pi_2]_{B_2} = B.

Since $[\pi_1]_{B_1} = [\pi_2]_{B_2}$ , we have $A = B$ .

Now, let $f: V_1 \to V_2$ be the linear map defined by $f(e_i) = f(e_i)$ for each $i$ . Then, we have

f(\pi_1(g) v) = f(g v) = g f(v) = \pi_2(g) f(v).

This shows that $f$ is an equivariant map.

We claim that $f$ is an isomorphism. To see this, let $v \in V_1$ be a vector and let $w = f(v) \in V_2$ be its image under $f$ . Then, we can write $v = \sum_{i=1}^n c_i e_i$ for some scalars $c_i$ . We have

f(v) = f\left(\sum_{i=1}^n c_i e_i\right) = \sum_{i=1}^n c_i f(e_i) = \sum_{i=1}^n c_i A e_i = A \left(\sum_{i=1}^n c_i e_i\right) = A v.

This shows that $f$ is an isomorphism.

Therefore, the representations $(\pi_1, V_1)$ and $(\pi_2, V_2)$ are equivalent.

Conclusion

In this article, we have shown that two representations are equivalent if and only if their matrices are equal in some bases. This result provides a more concrete and intuitive understanding of equivalence in representation theory. We have also provided a proof of this result, which involves showing that equivalence implies equal matrices and that equal matrices imply equivalence.

References

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

Glossary

Representation: A pair $(\pi, V)$ , where $V$ is a vector space and $\pi: G \to \text{GL}(V)$ is a homomorphism from $G$ to the general linear group of $V$ .
Equivariant map: A linear map $f: V_1 \to V_2$ such that $f(\pi_1(g)v) = \pi_2(g)f(v)$ for all $g \in G$ and $v \in V_1$ .
Equivalent representations: Two representations $(\pi_1, V_1)$ and $(\pi_2, V_2)$ are equivalent if there exists an equivariant map $f: V_1 \to V_2$ that is both injective and surjective.
Matrix of a representation: The matrix of a representation
Q&A: Representation Theory and Equivalence of Representations ===========================================================

Introduction

In our previous article, we showed that two representations are equivalent if and only if their matrices are equal in some bases. This result provides a more concrete and intuitive understanding of equivalence in representation theory. In this article, we will answer some frequently asked questions about representation theory and equivalence of representations.

Q: What is a representation?

Q: What is an equivariant map?

Q: What is equivalence of representations?

Q: How do I determine if two representations are equivalent?

To determine if two representations are equivalent, you can use the following steps:

Find the matrices of the two representations with respect to some bases.
Check if the matrices are equal.
If the matrices are equal, then the representations are equivalent.

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

Here are some common mistakes to avoid when working with representations:

Not checking if the map is equivariant: Make sure that the map you are using is equivariant before claiming that two representations are equivalent.
Not checking if the map is injective and surjective: Make sure that the map you are using is both injective and surjective before claiming that two representations are equivalent.
Not using the correct bases: Make sure that you are using the correct bases when finding the matrices of the representations.

Q: What are some applications of representation theory?

Representation theory has many applications in mathematics and physics, including:

Group theory: Representation theory is used to study the properties of groups and their actions on vector spaces.
Physics: Representation theory is used to study the symmetries of physical systems and to classify particles.
Computer science: Representation theory is used in computer science to study the properties of algorithms and data structures.

Q: What are some resources for learning more about representation theory?

Here are some resources for learning more about representation theory:

Books: "Representation theory: A first course" by Fulton and Harris, "Linear representations of finite groups" by Serre.
Online resources: Khan Academy, MIT OpenCourseWare.
Research papers: Search for research papers on arXiv or Google Scholar.

Conclusion

In this article, we have answered some frequently asked questions about representation theory and equivalence of representations. We hope that this article has been helpful in clarifying some of the concepts and ideas in representation theory. If you have any further questions, please don't hesitate to ask.

Glossary

Representation: A pair $(\pi, V)$ , where $V$ is a vector space and $\pi: G \to \text{GL}(V)$ is a homomorphism from $G$ to the general linear group of $V$ .
Equivariant map: A linear map $f: V_1 \to V_2$ such that $f(\pi_1(g)v) = \pi_2(g)f(v)$ for all $g \in G$ and $v \in V_1$ .
Equivalent representations: Two representations $(\pi_1, V_1)$ and $(\pi_2, V_2)$ are equivalent if there exists an equivariant map $f: V_1 \to V_2$ that is both injective and surjective.
Matrix of a representation: The matrix of a representation with respect to some basis.

Glossary

Representation: A pair $(\pi, V)$ , where $V$ is a vector space and $\pi: G \to \text{GL}(V)$ is a homomorphism from $G$ to the general linear group of $V$ .
Equivariant map: A linear map $f: V_1 \to V_2$ such that $f(\pi_1(g)v) = \pi_2(g)f(v)$ for all $g \in G$ and $v \in V_1$ .
Equivalent representations: Two representations $(\pi_1, V_1)$ and $(\pi_2, V_2)$ are equivalent if there exists an equivariant map $f: V_1 \to V_2$ that is both injective and surjective.
Matrix of a representation: The matrix of a representation with respect to some basis.