Problem 6 - Algebra

Question

Dihedral group \(D_{2 m^{-}}\) There is a group of order \(2 n\) (n even integer \(\geqq 2\) ) generated by two elements \(\sigma, \tau\) such that $$ \sigma^{n}=1, \tau^{2}=1, \text { and } \tau \sigma \tau=\sigma^{-1} . $$ It is called the dihedral group. (a) Show that there are four representations of dimension 1 , obtained by the four possible values \(\pm 1\) for \(\sigma\) and \(\tau\). (b) Let \(C_{n}\) be the cyclic subgroup of \(D_{2 n}\) generated by \(\sigma\). For each integer \(r=0, \ldots, n-1\) let \(\psi\), be the character of \(C_{n}\) such that $$ \psi_{r}(\sigma)=\zeta^{r} \quad(\zeta=\text { prim. } n \text { -th root of unity }) $$ Let \(\chi_{r}\), be the induced character. Show that \(\chi_{r}=\chi_{n-r \cdot}\) (c) Show that for \(0

Step-by-Step Solution

Answer

We found four one-dimensional representations of the dihedral group \(D_{2n}\), obtained by the four possible values \(\pm 1\) for \(\sigma\) and \(\tau\). For each integer \(r=0, \ldots, n-1\), we proved the relation between induced characters \(\chi_r\) derived from a cyclic subgroup of \(D_{2n}\), showing that \(\chi_r = \chi_{n-r}\). We also showed that for \(0

Step 1: Show the Four One-Dimensional Representations

Recall that a one-dimensional representation of a group \(G\) is a homomorphism \(\rho: G \to \mathbb{C}^{\times}\), where \(\mathbb{C}^{\times}\) is the group of nonzero complex numbers under multiplication. We are given that \(\sigma^n = 1, \tau^2 = 1\), and \(\tau\sigma\tau = \sigma^{-1}\), which are the defining relations of the dihedral group \(D_{2n}\). Thus, we can evaluate the homomorphism for the elements \(\sigma\) and \(\tau\): $$ \rho(\sigma^n) = 1 \quad\text{and}\quad \rho(\tau^2) = 1, $$ which implies $$ \rho(\sigma)^n = 1 \quad\text{and}\quad \rho(\tau)^2 = 1. $$ Since the representation is one-dimensional, \(\rho(\sigma), \rho(\tau) \in \mathbb{C}^{\times}\) must be scalar quantities that satisfy the conditions above. For both situations, the scalar must be either \(1\) or \(-1\). This gives us four combinations of \((\rho(\sigma), \rho(\tau))\): \((1, 1), (1, -1), (-1, 1),\) and \((-1, -1)\). These combinations correspond to the four one-dimensional representations of the dihedral group \(D_{2n}\).

Step 2: Induced Characters \(\chi_r\) Relation

For each integer \(r = 0, \ldots, n - 1\), let \(\psi_r(\sigma) = \zeta^r\), where \(\zeta\) is a primitive \(n\)-th root of unity. Let \(\chi_r\) be the character induced by \(\psi_r\). To show that \(\chi_r = \chi_{n-r}\), we will compute \(\chi_r(\sigma)\) and \(\chi_{n-r}(\sigma)\) and then compare their values. Note that \(\chi_r(\sigma) = \sum_{g \in G} \psi_r^G(g^{-1}\sigma g)\), where \(\psi_r^G\) denotes the character extending \(\psi_r\) to the whole group \(G\) (in this case, \(D_{2n}\)). For calculating \(\chi_r(\sigma)\), we need to take into account two cases: 1. When \(g = \sigma^k\) with \(0 \leq k \leq n-1\), we compute: $$ \psi_r^G(g^{-1}\sigma g) = \psi_r^C(\sigma^{k}\sigma\sigma^{-k}) = \psi_r(\sigma) = \zeta^r. $$ 2. When \(g = \sigma^k\tau\) with \(0 \leq k \leq n-1\), we compute (using the fact that \(\tau\sigma\tau = \sigma^{-1}\)): $$ \psi_r^G(g^{-1}\sigma g) = \psi_r^C(\tau\sigma^k\sigma\sigma^{-k}\tau) = \psi_r^C(\tau\sigma\tau) = \psi_r(\tau^2\sigma^{n-r}) = \zeta^{n-r}. $$ Summing up the values for both cases, we obtain: $$ \chi_r(\sigma) = n\zeta^r + 0. $$ Similarly, we can compute \(\chi_{n-r}(\sigma)\). By evaluating both cases, we ultimately obtain: $$ \chi_{n-r}(\sigma) = n\zeta^r + 0, $$ which proves that \(\chi_r = \chi_{n-r}\).

Step 3: Induced Character \(\chi_r\) Properties

We have to show that for \(0 < r < \frac{n}{2}\), the induced character \(\chi_r\) is simple, of dimension 2, and that one gets thereby \(\left(\frac{n}{2}-1\right)\) distinct characters of dimension 2. For \(0

Step 4: All Simple Characters of \(D_{2n}\)

We have found four simple characters of dimension 1 and \(\left(\frac{n}{2}-1\right)\) distinct simple characters of dimension 2. These characters comprise the following dimensions: $$ 4\cdot1 + \left(\frac{n}{2}-1\right)\cdot 2 = 4 + n - 2 = 2n. $$ As the number of dimensions found is equal to the order of \(D_{2n}\), these simple characters account for all simple characters of the dihedral group \(D_{2n}\).

Key Concepts

Group Theory

Group theory is a fascinating branch of mathematics focused on sets equipped with a binary operation that combines any two elements to form a third element. A group is defined by four key properties: closure, associativity, identity, and inverses. For example, the Dihedral Group, denoted as \(D_{2n}\), is the group of symmetries of a regular polygon with \(n\) sides, which can include both rotations and reflections. This group has an order of \(2n\), indicating that it has \(2n\) elements such as rotations and reflections.
In this exercise, we investigate the dihedral group through its generator elements \(\sigma\) and \(\tau\), where \(\sigma\) represents rotation and \(\tau\) represents reflection. The relations \(\sigma^n = 1\), \(\tau^2 = 1\), and \(\tau\sigma\tau = \sigma^{-1}\) are critical in defining the group's structure. Understanding these relations allows us to explore how elements interact within the group, unveiling their symmetry properties.

Representation Theory

Representation theory provides a way to abstractly describe groups by representing their elements as linear transformations of vector spaces. A representation of a group is a homomorphism from the group to the general linear group of matrices, ultimately making it easier to manipulate and understand the group's structure.
In the dihedral group exercise, we identify one-dimensional representations, where each group element corresponds to a scalar. By solving the homomorphism equation \(\rho(\sigma^n) = 1\) and \(\rho(\tau^2) = 1\), possible values of the representations are \(\pm 1\) for both \(\sigma\) and \(\tau\). These allowed scalar values grant insight into the simplicity and subtleties of group structure.

Cyclic Subgroup

A cyclic subgroup is a subgroup generated by a single element, where every element in the subgroup can be expressed as a power of this generator. In the context of the dihedral group \(D_{2n}\), the cyclic subgroup \(C_n\) is generated by the element \(\sigma\) representing rotation. The importance of cyclic subgroups in group theory is that they can simplify the understanding of the overall group's structure.
This exercise also explores characters \(\psi_r\) of the cyclic subgroup, rooted in the power properties of a primitive \(n\)-th root of unity \(\zeta\). The characters provide crucial perspectives when computing induced characters and examining the relationships and interactions among group elements.

Induced Character

Induced characters come into play when expanding a character of a subgroup to the entire group. They demonstrate how characters of subgroups carry into the full group structure, providing key insights into the overall representations.
In our example, the character \(\psi_r\) from the cyclic subgroup is extended to form an induced character \(\chi_r\) over the dihedral group. This process is fascinating as it allows us to see patterns like \(\chi_r = \chi_{n-r}\). Furthermore, these induced characters of dimension 2 demonstrate simplicity and distinctiveness in certain ranges of \(r\), illustrating underlying symmetry and balance.

Primitive Root of Unity

A primitive root of unity \(\zeta\) is a complex number whose powers generate distinct roots of unity. It plays a significant role in group theory and number theory, providing a pathway to construct complex roots and study periodic functions.
In this exercise, the \(n\)-th primitive root of unity \(\zeta\) is employed to define characters \(\psi_r(\sigma) = \zeta^r\). It's used as a tool to dissect and understand the induced characters by leveraging its repetition (periodicity) properties. Utilizing primitive roots of unity emboldens calculations and manipulations within the cyclic subgroup and by extension, the whole dihedral group.

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Problem 5

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Problem 7

Other exercises in this chapter

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Key Concepts

Group Theory

Group theory is a fascinating branch of mathematics focused on sets equipped with a binary operation that combines any two elements to form a third element. A group is defined by four key properties: closure, associativity, identity, and inverses. For example, the Dihedral Group, denoted as \(D_{2n}\), is the group of symmetries of a regular polygon with \(n\) sides, which can include both rotations and reflections. This group has an order of \(2n\), indicating that it has \(2n\) elements such as rotations and reflections.
In this exercise, we investigate the dihedral group through its generator elements \(\sigma\) and \(\tau\), where \(\sigma\) represents rotation and \(\tau\) represents reflection. The relations \(\sigma^n = 1\), \(\tau^2 = 1\), and \(\tau\sigma\tau = \sigma^{-1}\) are critical in defining the group's structure. Understanding these relations allows us to explore how elements interact within the group, unveiling their symmetry properties.

Representation Theory

Representation theory provides a way to abstractly describe groups by representing their elements as linear transformations of vector spaces. A representation of a group is a homomorphism from the group to the general linear group of matrices, ultimately making it easier to manipulate and understand the group's structure.
In the dihedral group exercise, we identify one-dimensional representations, where each group element corresponds to a scalar. By solving the homomorphism equation \(\rho(\sigma^n) = 1\) and \(\rho(\tau^2) = 1\), possible values of the representations are \(\pm 1\) for both \(\sigma\) and \(\tau\). These allowed scalar values grant insight into the simplicity and subtleties of group structure.

Cyclic Subgroup

A cyclic subgroup is a subgroup generated by a single element, where every element in the subgroup can be expressed as a power of this generator. In the context of the dihedral group \(D_{2n}\), the cyclic subgroup \(C_n\) is generated by the element \(\sigma\) representing rotation. The importance of cyclic subgroups in group theory is that they can simplify the understanding of the overall group's structure.
This exercise also explores characters \(\psi_r\) of the cyclic subgroup, rooted in the power properties of a primitive \(n\)-th root of unity \(\zeta\). The characters provide crucial perspectives when computing induced characters and examining the relationships and interactions among group elements.

Induced Character

Induced characters come into play when expanding a character of a subgroup to the entire group. They demonstrate how characters of subgroups carry into the full group structure, providing key insights into the overall representations.
In our example, the character \(\psi_r\) from the cyclic subgroup is extended to form an induced character \(\chi_r\) over the dihedral group. This process is fascinating as it allows us to see patterns like \(\chi_r = \chi_{n-r}\). Furthermore, these induced characters of dimension 2 demonstrate simplicity and distinctiveness in certain ranges of \(r\), illustrating underlying symmetry and balance.

Primitive Root of Unity

A primitive root of unity \(\zeta\) is a complex number whose powers generate distinct roots of unity. It plays a significant role in group theory and number theory, providing a pathway to construct complex roots and study periodic functions.
In this exercise, the \(n\)-th primitive root of unity \(\zeta\) is employed to define characters \(\psi_r(\sigma) = \zeta^r\). It's used as a tool to dissect and understand the induced characters by leveraging its repetition (periodicity) properties. Utilizing primitive roots of unity emboldens calculations and manipulations within the cyclic subgroup and by extension, the whole dihedral group.