Standard representation of symmetric group:S3

This article describes a particular irreducible linear representation for the following group: symmetric group:S3. The representation is unique up to equivalence of linear representations and is irreducible, at least over its original field of definition in characteristic zero. The representation may also be definable over other characteristics by reducing the matrices modulo that characteristic, though it may behave somewhat differently in these characteristics.
For more on the linear representation theory of the group, see linear representation theory of symmetric group:S3.

This article discusses a two-dimensional faithful irreducible representation of symmetric group:S3, called the standard representation since it belongs to the family of standard representations of symmetric groups.

Summary

Item	Value
Degree of representation	2
Kernel of representation	trivial subgroup, i.e., it is a faithful linear representation in all characteristics.
Quotient on which it descends to a faithful linear representation	symmetric group:S3
Size of equivalence class under automorphisms	1
Size of equivalence class under Galois automorphisms	1, because the representation is realized over $\mathbb {Z}$ in characteristic zero and more generally is realized over the prime subfield in any characteristic.
Schur index	1 in all characteristics
Bad characteristics	3: in characteristic 3, the representation can be defined and is indecomposable but not irreducible (more below).

Table with representation information

The representation can be defined in many equivalent ways. Note that the first two description columns show that the representation can be realized over $\mathbb {Z}$. In particular, this allows us to define it over any field and more generally over any unital ring by composing with the homomorphism from $\mathbb {Z}$ to that ring. In characteristic 3, however, the representation is not irreducible (as discussed later).

Element	Matrix for standard representation with basis $e_{1}-e_{2}$, $e_{2}-e_{3}$	Matrix for standard representation viewed as quotient with basis ${\overline {e_{1}}},{\overline {e_{2}}}$	Matrix for real representation as dihedral group	Matrix for complex representation as dihedral group	Characteristic polynomial	Minimal polynomial	Trace, character value	Determinant
Identity element	${\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}$	${\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}$	${\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}$	${\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}$	$(t-1)^{2}$	$t-1$	2	1
$(1,2,3)$	${\begin{pmatrix}0&-1\\1&-1\\\end{pmatrix}}$	${\begin{pmatrix}0&-1\\1&-1\\\end{pmatrix}}$	${\begin{pmatrix}-1/2&-{\sqrt {3}}/2\\{\sqrt {3}}/2&-1/2\end{pmatrix}}$	${\begin{pmatrix}e^{2\pi i/3}&0\\0&e^{-2\pi i/3}\end{pmatrix}}$	$t^{2}+t+1$	$t^{2}+t+1$	-1	1
$(1,3,2)$	${\begin{pmatrix}-1&1\\-1&0\\\end{pmatrix}}$	${\begin{pmatrix}-1&1\\-1&0\\\end{pmatrix}}$	${\begin{pmatrix}-1/2&{\sqrt {3}}/2\\-{\sqrt {3}}/2&-1/2\end{pmatrix}}$	${\begin{pmatrix}e^{-2\pi i/3}&0\\0&e^{2\pi i/3}\end{pmatrix}}$	$t^{2}+t+1$	$t^{2}+t+1$	-1	1
$(1,2)$	${\begin{pmatrix}-1&1\\0&1\\\end{pmatrix}}$	${\begin{pmatrix}0&1\\1&0\\\end{pmatrix}}$	${\begin{pmatrix}-1/2&{\sqrt {3}}/2\\{\sqrt {3}}/2&1/2\end{pmatrix}}$	${\begin{pmatrix}0&1\\1&0\\\end{pmatrix}}$	$t^{2}-1$	$t^{2}-1$	0	-1
$(2,3)$	${\begin{pmatrix}1&0\\1&-1\\\end{pmatrix}}$	${\begin{pmatrix}1&-1\\0&-1\end{pmatrix}}$	${\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$	${\begin{pmatrix}0&e^{-2\pi i/3}\\e^{2\pi i/3}&0\end{pmatrix}}$	$t^{2}-1$	$t^{2}-1$	0	-1
$(1,3)$	${\begin{pmatrix}0&-1\\-1&0\\\end{pmatrix}}$	${\begin{pmatrix}-1&0\\-1&1\\\end{pmatrix}}$	${\begin{pmatrix}-1/2&-{\sqrt {3}}/2\\-{\sqrt {3}}/2&1/2\end{pmatrix}}$	${\begin{pmatrix}0&e^{2\pi i/3}\\e^{-2\pi i/3}&0\end{pmatrix}}$	$t^{2}-1$	$t^{2}-1$	0	-1

Interpretations

Interpretation as symmetric group

This is a faithful two-dimensional representation. One way of obtaining this representation is as follows: consider a three-dimensional vector space with basis $e_{1},e_{2},e_{3}$. Let the symmetric group permute the basis vectors, and consider the induced action of the symmetric group on the vector space. This is a three-dimensional representation. Consider the two-dimensional subspace of all vectors of the form $x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}$ where $x_{1}+x_{2}+x_{3}=0$. When the characteristic of the field is not two or three, this is a faithful, irreducible, two-dimensional representation. Note that $e_{1}-e_{2}$ and $e_{2}-e_{3}$ can be taken as a basis for this, with $e_{3}-e_{1}$ being the negative of the sum of these.

Here are the details on how the matrices are computed.

Interpretation as dihedral group

Over the real numbers, this representation is conjugate to the representation as orthogonal matrices, where we view the symmetric group of degree three as the dihedral group acting on three elements. Here, the $3$-cycles act as rotations by multiples of $2\pi /3$, and the transpositions act as reflections about suitable axes.

Behavior in specific characteristics

The case of characteristic 3

In this case, the representation is indecomposable, but not irreducible. Here is an alternative perspective on this representation.
In characteristic three, the symmetric group is identified with the general affine group of degree one over the field of three elements. In other words, it is the semidirect product of the additive group of this field (a cyclic group of order three) and the multiplicative group of this field. Via the general fact that embeds a general affine group in a general linear group of one size higher, we get a faithful representation of the symmetric group on three elements in the general linear group of degree two over field:F3, i.e., in $GL(2,3)$.

Embeddings

Embeddings in general linear groups

For any field (and more generally, any commutative unital ring), this faithful irreducible representation defines an embedding of symmetric group:S3 into the general linear group of degree two over the field or ring. Further, because this is the only faithful representation of degree two (up to equivalence) the image of symmetric group:S3 is an isomorph-conjugate subgroup in the target group.

Further, the nature of the representation makes it clear that the representation in fact descends to an embedding of symmetric group:S3 in the projective general linear group of degree two over the field or commutative unital ring. Note, however, that it is not necessary that this embedding be as an isomorph-conjugate subgroup, because $S_{3}$ may have other projective representations. For more, see projective representation theory of symmetric group:S3.

We consider the special case of finite prime fields and elaborate on the embeddings below:

Field size	Field information	General linear group of degree two	Projective general linear group of degree two	Embedding in general linear group of degree two	Embedding in projective general linear group of degree two	Isomorph-conjugate subgroup in projective general linear group?
2	field:F2	symmetric group:S3	symmetric group:S3	whole group	whole group	Yes
3	field:F3	general linear group:GL(2,3)	symmetric group:S4	S3 in GL(2,3)	S3 in S4	Yes
4	field:F4	general linear group:GL(2,4)	alternating group:A5	not available	twisted S3 in A5	Yes
5	field:F5	general linear group:GL(2,5)	symmetric group:S5	S3 in GL(2,5)	S3 in S5	No -- there is twisted S3 in S5 which arises from a projective representation
7	field:F7	general linear group:GL(2,7)	projective general linear group:PGL(2,7)	S3 in GL(2,7)	?	?