representation theory of the dihedral groups

Semi-magic Matrices for Dihedral Groups

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First Online: 01 January 2023
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representation theory of the dihedral groups

Robert W. Donley 2

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 395))

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Combinatorial and Additive Number Theory, New York Number Theory Seminar

After reviewing the group structure and representation theory for the dihedral group $D_{2n},$ we consider an intertwining operator $\varPhi _\rho $ from the group algebra $\mathbb {C}[D_{2n}]$ into a corresponding space of semi-magic matrices. From this intertwining operator, one obtains the generating function for enumerating the associated semi-magic squares with fixed line sum and an algebra extending the circulant matrices. While this work complements the approach to $D_{2n}$ through permutation polytopes, we use only methods from representation theory.

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On the Commuting Graph of Semidihedral Group

Homometry in the Dihedral Groups: Lifting Sets from $$ \mathbb {Z}_{n}$$ to $$ D_{n}$$

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Robert W. Donley

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Donley, R.W. (2022). Semi-magic Matrices for Dihedral Groups. In: Nathanson, M.B. (eds) Combinatorial and Additive Number Theory V. CANT 2021. Springer Proceedings in Mathematics & Statistics, vol 395. Springer, Cham. https://doi.org/10.1007/978-3-031-10796-2_6

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4.2: Dihedral Groups

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We can think of finite cyclic groups as groups that describe rotational symmetry. In particular, $R_n$ is the group of rotational symmetries of a regular $n$ -gon. Dihedral groups are those groups that describe both rotational and reflectional symmetry of regular $n$ -gons.

Definition: Dihedral Group

For $n\geq 3$ , the dihedral group $D_n$ is defined to be the group consisting of the symmetry actions of a regular $n$ -gon, where the operation is composition of actions.

For example, as we’ve seen, $D_3$ and $D_4$ are the symmetry groups of equilateral triangles and squares, respectively. The symmetry group of a regular pentagon is denoted by $D_5$ . It is a well-known fact from geometry that the composition of two reflections in the plane is a rotation by twice the angle between the reflecting lines.

Theorem $\PageIndex{1}$

The group $D_n$ is a non-abelian group of order $2n$ .

Theorem $\PageIndex{2}$: Generators $D_n$

Fix $n\geq 3$ and consider $D_n$ . Let $r$ be rotation clockwise by $360^{\circ}/n$ and let $s$ and $s'$ be any two adjacent reflections of a regular $n$ -gon. Then

$D_n=\langle r,s\rangle =\{\underbrace{e,r,r^2,\ldots, r^{n-1}}_{\text{rotations}},\underbrace{s,sr,sr^2,\ldots,sr^{n-1}}_{\text{reflections}}\}$ and
$D_n=\langle s,s'\rangle = \text{all possible products of }s\text{ and }s'$ .

The next result is an obvious corollary of Theorem $\PageIndex{2}$.

Corollary $\PageIndex{1}$

For $n\geq 3$ , $R_n\leq D_n$ .

The following theorem generalizes many of the relations we have witnessed in the Cayley diagrams for the dihedral groups $D_3$ and $D_4$ .

Exercise $\PageIndex{1}$

Fix $n\geq 3$ and consider $D_n$ . Let $r$ be rotation clockwise by $360^{\circ}/n$ and let $s$ and $s'$ be any two adjacent reflections of a regular $n$ -gon. Then the following relations hold.

$r^n = s^2 = (s')^2 =e$ ,
$r^{-k} = r^{n-k}$ (special case: $r^{-1}=r^{n-1}$ ),
$sr^k=r^{n-k}s$ (special case: $sr=r^{n-1}s$ ),
$\underbrace{ss's\cdots}_{n\text{ factors}}=\underbrace{s'ss'\cdots}_{n\text{ factors}}$ .

Exercise $\PageIndex{2}$

From Theorem $\PageIndex{2}$, we know \[D_n=\langle r,s\rangle =\{\underbrace{e,r,r^2,\ldots, r^{n-1}}_{\text{rotations}},\underbrace{s,sr,sr^2,\ldots,sr^{n-1}}_{\text{reflections}}\}.\] If you were to create the group table for $D_n$ so that the rows and columns of the table were labeled by $e,r,r^2,\ldots, r^{n-1},s,sr,sr^2,\ldots,sr^{n-1}$ (in exactly that order), do any patterns arise? Where are the rotations? Where are the reflections?

Exercise $\PageIndex{3}$

What does the Cayley diagram for $D_n$ look like if we use $\{r,s\}$ as the generating set? What if we use $\{s,s'\}$ as the generating set?

Dihedral Group D_3

To find the irreducible representation, note that there are three conjugacy classes . The fifth rule of irreducible representations requires that there be three irreducible representations, and the second rule requires that

so it must be true that

By rule 6, we can let the first representation have all 1s.

Using group rule 1, we see that

so the final representation for 1 has group character 2. Orthogonality with the first two representations ( group rule 3) then yields the following constraints:

Since there are only three conjugacy classes , this table is conventionally written simply as

Writing the irreducible representations in matrix form then yields

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Weisstein, Eric W. "Dihedral Group D_3." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/DihedralGroupD3.html

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Title: looking for a refined monster.

Abstract: We discuss some categorical aspects of the objects that appear in the construction of the Monster and other sporadic simple groups. We define the basic representation of the categorical torus $\mathcal T$ classified by an even symmetric bilinear form $I$ and of the semi-direct product of $\mathcal T$ with its canonical involution. We compute the centraliser of the basic representation of $\mathcal T\rtimes\{\pm1\}$ and find it to be a categorical extension of the extraspecial $2$-group with commutator $I\mod 2$. We study the inertia groupoid of a categorical torus and find that it is given by the torsor of the topological Looijenga line bundle, so that $2$-class functions on $\mathcal T$ are canonically theta-functions. We discuss how discontinuity of the categorical character in our formalism means that the character of the basic representation fails to be a categorical class function. We compute the automorphisms of $\mathcal T$ and of $\mathcal T\rtimes\{\pm1\}$ and relate these to the Conway groups.

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