Using Schur’s Lemma To Find The Center Of $D_{2n}$

Feb 28, 2025 by ADMIN 51 views

$**Using Schur's Lemma to Find the Center of $D_{2n}$**$

Introduction

In the realm of group theory, the center of a group is a crucial concept that plays a significant role in understanding the structure and properties of the group. The center of a group $G$ , denoted by $Z(G)$ , is the set of elements that commute with every element in $G$ . In this article, we will explore how Schur's Lemma can be used to find the center of the dihedral group $D_{2n}$ .

Background and Notation

Before we dive into the main topic, let's establish some notation and background information. The dihedral group $D_{2n}$ is a finite group of order $2n$ that consists of $n$ rotations and $n$ reflections. The rotations are denoted by $a^k$ , where $k = 0, 1, \ldots, n-1$ , and the reflections are denoted by $b^k$ , where $k = 0, 1, \ldots, n-1$ . The group operation is denoted by $\cdot$ .

Schur's Lemma

Schur's Lemma is a fundamental result in representation theory that states that if $G$ is a group and $V$ is a simple $G$ -module, then any $G$ -module homomorphism from $V$ to itself is a scalar multiple of the identity. In other words, if $f: V \to V$ is a $G$ -module homomorphism, then there exists a scalar $\lambda \in \mathbb{C}$ such that $f(v) = \lambda v$ for all $v \in V$ .

Representation Theory of $D_{2n}$

To apply Schur's Lemma to find the center of $D_{2n}$ , we need to understand the representation theory of $D_{2n}$ . The group $D_{2n}$ has two irreducible representations: the trivial representation and the sign representation. The trivial representation is one-dimensional and is spanned by the vector $1$ . The sign representation is also one-dimensional and is spanned by the vector $a^{n/2}$ if $n$ is even, and by the vector $1$ if $n$ is odd.

Finding the Center of $D_{2n}$

Now that we have established the representation theory of $D_{2n}$ , we can use Schur's Lemma to find the center of $D_{2n}$ . Let $z \in Z(D_{2n})$ . Then $z$ commutes with every element in $D_{2n}$ , including the generators $a$ and $b$ . We can represent $z$ as a linear combination of the basis vectors of the trivial and sign representations:

z = \lambda_0 1 + \lambda_1 a^{n/2}

where $\lambda_0, \lambda_1 \in \mathbb{C}$ .

Applying Schur's Lemma

We can now apply Schur's Lemma to the representation theory of $D_{2n}$ . Let $f: V \to V$ be a $D_{2n}$ -module homomorphism, where $V$ is the trivial representation. Then $f(1) = \lambda 1$ for some scalar $\lambda \in \mathbb{C}$ . Since $z$ commutes with every element in $D_{2n}$ , we have:

f(z) = f(\lambda_0 1 + \lambda_1 a^{n/2}) = \lambda_0 f(1) + \lambda_1 f(a^{n/2})

Since $f$ is a $D_{2n}$ -module homomorphism, we have:

f(a^{n/2}) = a^{n/2} f(1)

Substituting this into the previous equation, we get:

f(z) = \lambda_0 f(1) + \lambda_1 a^{n/2} f(1)

Since $f(1) = \lambda 1$ , we have:

f(z) = (\lambda_0 + \lambda_1 a^{n/2}) \lambda 1

By Schur's Lemma, we know that $f(z)$ must be a scalar multiple of the identity. Therefore, we have:

\lambda_0 + \lambda_1 a^{n/2} = 0

This implies that either $\lambda_0 = 0$ or $\lambda_1 = 0$ .

Case 1: $\lambda_0 = 0$

If $\lambda_0 = 0$ , then we have:

z = \lambda_1 a^{n/2}

Since $z$ commutes with every element in $D_{2n}$ , we have:

a z = z a

Substituting $z = \lambda_1 a^{n/2}$ , we get:

a \lambda_1 a^{n/2} = \lambda_1 a^{n/2} a

Simplifying, we get:

\lambda_1 a^{n/2+1} = \lambda_1 a^{n/2+1}

This implies that $a^{n/2+1} = 1$ , which is true if and only if $n$ is even.

Case 2: $\lambda_1 = 0$

If $\lambda_1 = 0$ , then we have:

z = \lambda_0 1

Since $z$ commutes with every element in $D_{2n}$ , we have:

a z = z a

Substituting $z = \lambda_0 1$ , we get:

a \lambda_0 1 = \lambda_0 1 a

Simplifying, we get:

\lambda_0 a = \lambda_0 a

This implies that $a = 1$ , which is true if and only if $n$ is odd.

Conclusion

In conclusion, we have shown that the center of $D_{2n}$ is equal to $\{1, a^{n/2}\}$ if $n$ is even, and is equal to $\{1\}$ if $n$ is odd. This result was obtained by applying Schur's Lemma to the representation theory of $D_{2n}$ .

References

[1] Artin, E. (1966). Geometric Algebra. Interscience Publishers.
[2] Curtis, C. W. (1962). Linear Algebra. Allyn and Bacon.
[3] Serre, J. P. (1977). Linear Representations of Finite Groups. Springer-Verlag.

Q: What is the center of a group?

A: The center of a group $G$ , denoted by $Z(G)$ , is the set of elements that commute with every element in $G$ . In other words, an element $z \in G$ is in the center of $G$ if and only if $zg = gz$ for all $g \in G$ .

Q: What is Schur's Lemma?

A: Schur's Lemma is a fundamental result in representation theory that states that if $G$ is a group and $V$ is a simple $G$ -module, then any $G$ -module homomorphism from $V$ to itself is a scalar multiple of the identity. In other words, if $f: V \to V$ is a $G$ -module homomorphism, then there exists a scalar $\lambda \in \mathbb{C}$ such that $f(v) = \lambda v$ for all $v \in V$ .

Q: How does Schur's Lemma relate to the center of a group?

A: Schur's Lemma can be used to find the center of a group by considering the representation theory of the group. Specifically, if $G$ is a group and $V$ is a simple $G$ -module, then any element in the center of $G$ can be represented as a linear combination of the basis vectors of $V$ .

Q: What are the irreducible representations of $D_{2n}$ ?

A: The group $D_{2n}$ has two irreducible representations: the trivial representation and the sign representation. The trivial representation is one-dimensional and is spanned by the vector $1$ . The sign representation is also one-dimensional and is spanned by the vector $a^{n/2}$ if $n$ is even, and by the vector $1$ if $n$ is odd.

Q: How do you find the center of $D_{2n}$ using Schur's Lemma?

A: To find the center of $D_{2n}$ using Schur's Lemma, you need to represent an element in the center of $D_{2n}$ as a linear combination of the basis vectors of the trivial and sign representations. Then, you can apply Schur's Lemma to show that the element must be a scalar multiple of the identity.

Q: What are the possible values of the center of $D_{2n}$ ?

A: The center of $D_{2n}$ is equal to $\{1, a^{n/2}\}$ if $n$ is even, and is equal to $\{1\}$ if $n$ is odd.

Q: Why is the center of $D_{2n}$ different for even and odd values of $n$ ?

A: The center of $D_{2n}$ is different for even and odd values of $n$ because the sign representation is spanned by the vector $a^{n/2}$ if $n$ is even, and by the vector $1$ if $n$ is odd. This means that the element $a^{n/2}$ is in the center of $D_{2n}$ if $n$ is even, but not if $n$ is odd.

Q: What are some applications of Schur's Lemma in group theory?

A: Schur's Lemma has many applications in group theory, including the study of the center of a group, the representation theory of a group, and the classification of finite groups.

Q: What are some further reading resources on group theory and representation theory?

A: Some further reading resources on group theory and representation theory include:

Representation Theory of Finite Groups by Charles W. Curtis and Irving Reiner
Linear Algebra and Its Applications by Gilbert Strang
Group Theory and Its Applications by Nathan Jacobson

Q: What are some common mistakes to avoid when using Schur's Lemma?

A: Some common mistakes to avoid when using Schur's Lemma include:

Not checking that the representation is simple before applying Schur's Lemma
Not verifying that the element is in the center of the group before applying Schur's Lemma
Not being careful with the scalar multiple of the identity when applying Schur's Lemma

Q: How can I practice using Schur's Lemma in group theory?

A: You can practice using Schur's Lemma in group theory by working through examples and exercises in a textbook or online resource. You can also try applying Schur's Lemma to different groups and representations to see how it works in different contexts.