Representations of Groups from Geometric Methods

Adam WoodSummer 2018

In this note, we connect representations of finite groups to geometric methods. More specifically, we lookat three examples; representations of symmetric groups of order 12 and 24 as well as the dihedral group oforder 8 over C. Denote the symmetric groups by S3 and S4. Denote the dihedral group of order 8 by D4.We realize S3 as the group of symmetries of an equilateral triangle, we realize D4 as the group of symmetriesof a square, and we realize S4 as the rotational group of a cube. We then use these realizations to findrepresentations of the various groups.

S3

We realize S3 as the group of symmetries of an equilateral triangle and find a representation from the actionof S3 on the triangle. Label the vertices of an equilateral triangle as follows.

3

1

2

(0,0)

Then, S3 acts on the vertices of the triangle by permuting the indices. The representation will come fromthe transformation matrix describing the motion of the triangle under the action of an element in S3. Recall

that a rotation by θ is given by the matrix

(cos θ − sin θsin θ cos θ

)and a reflection about a line at angle θ with the

x-axis is given by the matrix

(cos 2θ sin 2θsin 2θ − cos 2θ

)The table below summarizes the elements corresponding

to the symmetries of the triangle and the transformation matrices in R2.

Action Resulting Triangle Element of S3 Transformation Matrix

None

3

1

2 (1)

(1 00 1

)

1

Rotation by 120◦

2

3

1 (1 3 2)

(− 1

2 −√32√

32 − 1

2

)

Rotation by 240◦

1

2

3 (1 2 3)

(− 1

2

√32

−√32 − 1

2

)

Reflection along line from 3

3

2

1 (1 2)

(1 00 −1

)

Reflection along line from 1

1

3

2 (1 3)

(− 1

2 −√32

−√32

12

)

Reflection along line from 2

2

1

3 (2 3)

(− 1

2

√32√

32

12

)

Calculating the trace of these matrices gives the character of the corresponding representation of S3. Thecharacter, χ, is given by

(1) (1 2) (1 3) (2 3) (1 2 3) (1 3 2)χ 2 0 0 0 −1 −1

which agrees with the character of the standard representation of S3.

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D4

We will realize D4 as the group of symmetries of a square. Consider the presentation of D4 given by

D4 = 〈r, s | r4 = 1 = s2, rsr−1 = s〉

so that r corresponds to a rotation by π/2 and s corresponds to reflection about the y-axis. Label thevertices of the square centered at the origin as shown below. The table below shows how the elements of D4

correspond to rotations and reflections of the square. Also, the matrix giving the transformation in R2 isgiven as well.

Action Resulting Square Element in D4 Transformation Matrix

None

AB

C D1

(1 00 1

)

Rotation by π2

DA

B Cr

(0 −11 0

)

Rotation by π

CD

A Br2

(−1 00 −1

)

Rotation by 3π2

BC

D Ar3

(0 1−1 0

)

Reflection about y-axis

BA

D Cs

(−1 00 1

)

Reflection about y-axis and rotation by π2

BC

D Asr

(0 11 0

)3

Reflection about y-axis and rotation by π

CD

A Bsr2

(1 00 −1

)

Reflection about y-axis and rotation by 3π2

DA

B Csr3

(0 −1−1 0

)

This correspondence between elements of D4 and matrices gives a representation. Calculating the trace ofthe matrices, we can see that the character, χ, of this representation is given by

1 r r2 r3 s sr sr2 sr3

χ 2 0 −2 0 0 0 0 0

Comparing these character values with the character table for D4 given in [1, pg. 37] shows that this repre-sentation is irreducible.

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S4

We will realize S4 as the group of rotations of a cube. Number the vertices of the cube as below. Note thateach face has vertices with distinct labels 1, 2, 3, and 4.

Let S4 act on the cube by permuting the vertices. For example, (1 2) acts on the cube by changing bothvertices labeled 1 to 2 and by changing both vertices labeled 2 to 1. We first investigate the rotationalsymmetries of the cube preserving orientation. A cube can be rotated by 90, 180, and 270 degrees aboutthe x, y, and z axes as shown below in (a). Also, a cube can be rotated by 180 degrees about the six axesshown below in (b). Finally, a cube can be rotated by 120 and 240 degrees about the four long diagonals asshown below in (c).

(a) (b) (c)

These options, along with the identity, give 24 rotations of the cube. Then, each rotation of the cubepreserving orientation corresponds to an element in S4. The table below show the correspondence betweenrotations and elements in S4.

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RotationAngle

Rotation AxisElement in

S4

0◦ (1)

90◦ (1 2 4 3)

180◦ (1 4)(2 3)

270◦ (1 2 4 3)

90◦ (1 4 3 2)

180◦ (1 3)(2 4)

270◦ (1 2 3 4)

90◦ (1 3 2 4)

180◦ (1 2)(3 4)

270◦ (1 4 2 3)

180◦ (1 2)

180◦ (3 4)

180◦ (1 3)

RotationAngle

Rotation AxisElement in

S4

180◦ (2 4)

180◦ (2 3)

180◦ (1 4)

120◦ (2 4 3)

240◦ (2 3 4)

120◦ (1 4 3)

240◦ (1 3 4)

120◦ (1 4 2)

240◦ (1 2 4)

120◦ (1 3 2)

240◦ (1 2 3)

How does this relate to representation theory? We have realized S4 as the group of rotational symmetriesof a cube. Through this realization, S4 acts on the faces, edges, and vertices of a cube. Each of these actionhas a permutation representation. We will find the character of each permutation representation and thendecompose each into a direct sum of irreducible representations. For a permutation representation on afinite set X, the character of a group element σ is the number of elements of X that σ fixes. Note that acube has 6 faces, 12 edges, and 8 vertices. So, the permutation representations will give six, twelve, and

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eight dimensional representations, respectively. We are only concerned with the value of each character ona representative of each conjugacy class. Choose representatives, {(1), (1 2), (1 2 3), (1 2)(3 4), (1 2 3 4)},of each conjugacy class. Considering the correspondence of elements of S4 with rotations of the cube givenin the table above, we will describe which rotations fix faces, edges, or vertices. The identity, (1), fixes allface, edges, and vertices. The rotation by 180 degrees corresponding to (1 2) fixes no faces, two edges, andno vertices. The rotation by 240 degrees along the long diagonal between the vertices labeled 4 fixes nofaces, no edges, and two vertices. The rotations by 180 degrees along the y-axis and by 270 degrees alongthe x-axis corresponding to (1 2)(3 4) and (1 2 3 4), respectively, fix two faces, no edges, and no vertices.Therefore, we have the following values of the characters of the corresponding permutation representations.

(1) (1 2) (1 2 3) (1 2)(3 4) (1 2 3 4)χfaces 6 0 0 2 2χedges 12 2 0 0 0χvertices 8 0 2 0 0

The character table or S4 is given as follows. See [1, pg. 43] for a reference.

(1) (1 2) (1 2 3) (1 2)(3 4) (1 2 3 4)χ0 1 1 1 1 1χ1 1 −1 1 1 −1χ2 2 0 −1 2 0χ3 3 1 0 −1 −1χ4 3 −1 0 −1 1

By direct computation, one can compute (χfaces|χi), (χedges|χi), and (χvertices|χi) for all χi irreduciblecharacters of S4 (see the document on examples) to see that

χfaces = χ0 + χ2 + χ4

χedges = χ0 + χ2 + 2χ3 + χ4

χvertices = χ0 + χ1 + χ3 + χ4

As a last example, we see how the action of S4 on the cube as the group of rotational symmetries of the cubecorresponds to a representation. Recall that in R3, the rotation by an angle θ about a axis in the directionof a unit vector (x, y, z) is given by the matrix cos θ + x2(1− cos θ) xy(1− cos θ)− z sin θ xz(1− cos θ) + y sin θ

yx(1− cos θ) + z sin θ cos θ + y2(1− cos θ) yz(1− cos θ)− x sin θzx(1− cos θ)− y sin θ zy(1− cos θ) + x sin θ cos θ + z2(1− cos θ)

For each representative of a conjugacy class, we will find the matrix corresponding to the rotation of thecube to determine the character of the representation associated to this action. For (1), there is no rotation,so the matrix is 1 0 0

0 1 00 0 1

For (1 2), this element corresponds to rotation by 180◦ about the axis in the direction of the unit vector

(√22 , 0,

√22 ) and therefore the matrix is 0 0 1

0 −1 01 0 0

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For (1 2 3), this element corresponds to rotation by 240◦ about the axis in the direction of the unit vector

(−√33 ,√33 ,√33 ) and therefore the matrix is 0 1 0

0 0 11 0 0

For (1 2)(3 4), this element corresponds to rotation by 180◦ about the y-axis, that is, in the direction of theunit vector (0, 1, 0) and therefore the matrix is−1 0 0

0 1 00 0 −1

For (1 2 3 4), this element corresponds to rotation by 270◦ about the x-axis, that is, in the direction of theunit vector (1, 0, 0) and therefore the matrix is1 0 0

0 0 10 −1 0

Let χ be the character of the representation coming from the realization of S4 as the group of rotations ofthe cube. Calculating the traces of the above matrices, we have the following values for the character of therepresentatives of the conjugacy classes.

(1) (1 2) (1 2 3) (1 2)(3 4) (1 2 3 4)χ 3 1 0 −1 1

We can then see that this representation is isomorphic to the standard representation of S4.

References

[1] Jean-Pierre Serre. Linear Representations of Finite Groups. Springer-Verlag, New York, 1977.

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