Representations Of Z N \mathbb{Z}_n Z N ​ Over Q \mathbb{Q} Q

Introduction

In the realm of abstract algebra, representation theory plays a vital role in understanding the structure of groups and their properties. The study of representations of a group over a field involves finding homomorphisms from the group to the general linear group of the field. In this article, we will delve into the representations of $\mathbb{Z}_n$, the cyclic group of order $n$, over the field of rational numbers $\mathbb{Q}$.

Background

Before we dive into the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$, let's briefly review the known results for $\mathbb{Z}_n$ over $\mathbb{C}$. It is well-known that the irreducible representations of $\mathbb{Z}_n$ over $\mathbb{C}$ are the $n$ group homomorphisms $\mathbb{Z}_n \to \mathbb{C}$ sending $1$ to the roots of unity. There are $n$ of these homomorphisms, and they are all irreducible. This result is a fundamental consequence of the theory of finite fields and the properties of roots of unity.

Representations of $\mathbb{Z}_n$ over $\mathbb{Q}$

Now, let's turn our attention to the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$. We know that $\mathbb{Q}$ is a subfield of $\mathbb{C}$, and therefore, any representation of $\mathbb{Z}_n$ over $\mathbb{Q}$ can be viewed as a representation over $\mathbb{C}$. However, the converse is not necessarily true. A representation of $\mathbb{Z}_n$ over $\mathbb{C}$ may not be defined over $\mathbb{Q}$.

To understand the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$, we need to consider the properties of the group $\mathbb{Z}_n$ and the field $\mathbb{Q}$. The group $\mathbb{Z}_n$ is a cyclic group of order $n$, and it has a single generator, which we can denote by $1$. The group operation is addition modulo $n$. On the other hand, the field $\mathbb{Q}$ is a field of rational numbers, which consists of all fractions of the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b$ is non-zero.

Irreducible Representations

An irreducible representation of $\mathbb{Z}_n$ over $\mathbb{Q}$ is a homomorphism $\rho: \mathbb{Z}_n \to \text{GL}_k(\mathbb{Q})$ that cannot be decomposed into a direct sum of smaller representations. In other words, the representation $\rho$ is irreducible if and only if the only $\mathbb{Q}$-subspaces of the representation space $V$ that are invariant under the action of $\mathbb{Z}_n$ are $0$ and $V$ itself.

To find the irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$, we need to consider the possible values of $k$, the dimension of the representation space $V$. Since $\mathbb{Z}_n$ is a cyclic group of order $n$, we know that the order of any element in the group is a divisor of $n$. Therefore, the possible values of $k$ are the divisors of $n$.

The Case of $k=1$

Let's start by considering the case where $k=1$. In this case, the representation space $V$ is one-dimensional, and the representation $\rho$ is a homomorphism $\rho: \mathbb{Z}_n \to \mathbb{Q}^*$. Since $\mathbb{Q}^*$ is a multiplicative group, we can identify $\mathbb{Q}^*$ with the group of units of the ring of integers of $\mathbb{Q}$.

The homomorphisms $\rho: \mathbb{Z}_n \to \mathbb{Q}^*$ are precisely the $n$ group homomorphisms $\mathbb{Z}_n \to \mathbb{Q}^*$ sending $1$ to the $n$-th roots of unity. These homomorphisms are all irreducible, and they form a complete set of irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$.

The Case of $k>1$

Now, let's consider the case where $k>1$. In this case, the representation space $V$ is a vector space of dimension $k$ over $\mathbb{Q}$. The representation $\rho$ is a homomorphism $\rho: \mathbb{Z}_n \to \text{GL}_k(\mathbb{Q})$, and we need to find the possible values of $\rho(1)$.

Since $\rho(1)$ is an element of $\text{GL}_k(\mathbb{Q})$, we know that $\rho(1)$ is a $k \times k$ invertible matrix with entries in $\mathbb{Q}$. The matrix $\rho(1)$ must satisfy the condition that $\rho(1)^n = 1$, where $1$ is the identity matrix.

To find the possible values of $\rho(1)$, we need to consider the properties of the matrix $\rho(1)$. Since $\rho(1)$ is a $k \times k$ matrix with entries in $\mathbb{Q}$, we know that the characteristic polynomial of $\rho(1)$ is a polynomial with coefficients in $\mathbb{Q}$.

The characteristic polynomial of $\rho(1)$ must be a polynomial of degree $k$ with roots in $\mathbb{C}$. Since $\rho(1)^n = 1$, we know that the characteristic polynomial of $\rho(1)$ must be a polynomial that divides $x^n - 1$.

Conclusion

In this article, we have discussed the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$. We have shown that the irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are the $n$ group homomorphisms $\mathbb{Z}_n \to \mathbb{Q}^*$ sending $1$ to the $n$-th roots of unity. We have also considered the case where $k>1$, and we have shown that the possible values of $\rho(1)$ are the $k \times k$ invertible matrices with entries in $\mathbb{Q}$ that satisfy the condition that $\rho(1)^n = 1$.

Q: What is the significance of the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$?

A: The representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are significant because they provide a fundamental understanding of the structure of the group $\mathbb{Z}_n$ and its properties. The representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are also important in the study of abstract algebra and representation theory.

Q: What are the irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$?

A: The irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are the $n$ group homomorphisms $\mathbb{Z}_n \to \mathbb{Q}^*$ sending $1$ to the $n$-th roots of unity. These homomorphisms are all irreducible, and they form a complete set of irreducible representations of $\mathbb{Z}_n$ over $\mathbb{Q}$.

Q: What is the relationship between the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ and the representations of $\mathbb{Z}_n$ over $\mathbb{C}$?

A: The representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are closely related to the representations of $\mathbb{Z}_n$ over $\mathbb{C}$. In fact, any representation of $\mathbb{Z}_n$ over $\mathbb{Q}$ can be viewed as a representation over $\mathbb{C}$. However, the converse is not necessarily true. A representation of $\mathbb{Z}_n$ over $\mathbb{C}$ may not be defined over $\mathbb{Q}$.

Q: What are the possible values of $k$ for the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$?

A: The possible values of $k$ for the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are the divisors of $n$. This is because the order of any element in the group $\mathbb{Z}_n$ is a divisor of $n$.

Q: What is the relationship between the characteristic polynomial of $\rho(1)$ and the polynomial $x^n - 1$?

A: The characteristic polynomial of $\rho(1)$ must be a polynomial that divides $x^n - 1$. This is because $\rho(1)^n = 1$, and therefore, the characteristic polynomial of $\rho(1)$ must be a polynomial that has $1$ as a root.

Q: What are the implications of the results obtained in this article for the study of abstract algebra and representation theory?

A: The results obtained in this article have significant implications for the study of abstract algebra and representation theory. They provide a fundamental understanding of the structure of the group $\mathbb{Z}_n$ and its properties, and they shed light on the between the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ and the representations of $\mathbb{Z}_n$ over $\mathbb{C}$.

Q: What are the potential applications of the results obtained in this article?

A: The results obtained in this article have potential applications in a variety of fields, including cryptography, coding theory, and number theory. They can be used to develop new cryptographic protocols, to construct new error-correcting codes, and to study the properties of numbers and their relationships.

Q: What are the future directions for research in this area?

A: There are several future directions for research in this area. One potential direction is to study the representations of other groups over $\mathbb{Q}$ and to investigate their properties. Another potential direction is to apply the results obtained in this article to develop new cryptographic protocols and to construct new error-correcting codes.

Q: What are the challenges and limitations of the results obtained in this article?

A: One challenge of the results obtained in this article is that they are based on the assumption that the representations of $\mathbb{Z}_n$ over $\mathbb{Q}$ are irreducible. However, this assumption may not always be true, and therefore, the results obtained in this article may not always be applicable. Another challenge is that the results obtained in this article are based on the use of complex analysis and representation theory, and therefore, they may not be accessible to researchers who do not have a strong background in these areas.