# $$\mathbf{S}_3 = \mathbf{D}_3$$

This is the symmetric group on three things. It can be realized as the group of symmetries of an equilateral triangle (in its incarnation as $$\mathbf{D}_3$$).

### Character table

 S3 e C2 C3 notes # 1 3 2 |S3| = 6 A0 1 1 1 trivial rep A1 1 -1 1 alternating rep E 2 0 -1
• All the irreducible representations are absolutely irreducible and are defined over the integers Z.

### Representation ring

 ⊗ A0 A1 E A0 A0 A1 E A1 A1 A0 E E E E A0+A1+E

It can be seen from this that the representation ring for S3 satisfies $$R(S_3) \simeq \mathbb{Z}[X, Y] / \left<XY-Y,\,X^2-1,\,Y^2-X-Y-1\right>,$$ where X is the alternating representation A1, and Y is the 2-dimensional irreducible E.

### Permutation representations

• The permutation representation on 3 points (vertices of an equilateral triangle) is A0 + E
• The "orientation permutation" representation on the set of 3 edges of the triangle is A1 + E

### Burnside ring

#### Table of marks:

The rows are the orbit types, the columns are the subgroups. The entries represent #Fix(H, G/K) - the number of elements in the orbit G/K fixed by the subgroup H.

 1 Z2 Z3 S3 O4 = S3/1 6 - - - O3 = S3/Z2 3 1 - - O2 = S3/Z3 2 0 2 - O1 = S3/S3 1 1 1 1

#### Product structure in Burnside Ring Ω(G):

 X O1 O2 O3 O4 O1 O1 O2 O3 O4 O2 O2 2O2 O4 2O4 O3 O3 O4 O3+O4 3O4 O4 O4 2O4 3O4 6O4

#### Homomorphism β: Ω(G) → R(G)

• β(O1) = A0
• β(O2) = A0+A1
• β(O3) = A0+E
• β(O4) = A0+A1+2E

Note that this is not injective: indeed the kernel is generated by 2O1 - O2 - 2O3 + O4. (This is a 'virtual set' of cardinality 0, as it must be.)