# Linear representation theory of dihedral group:D8

View linear representation theory of particular groups | View other specific information about dihedral group:D8

We shall use the dihedral group with the following presentation (here, $e$ is used to denote the identity element):

$\langle a,x \mid a^4 = x^2 = e, xax^{-1} = a^{-1} \rangle$.

## Summary

Item Value
Degrees of irreducible representations over a splitting field (such as $\mathbb{C}$ or $\overline{\mathbb{Q}}$) 1,1,1,1,2
maximum: 2, lcm: 2, number: 5, sum of squares: 8
Schur index values of irreducible representations 1,1,1,1,1
Smallest ring of realization for all irreducible representations (characteristic zero) $\mathbb{Z}$; same as ring generated by character values
Minimal splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero) $\mathbb{Q}$ (hence, it is a rational representation group)
Same as field generated by character values, because all Schur index values are 1.
Condition for being a splitting field for this group Any field of characteristic not two is a splitting field.
Minimal splitting field in characteristic $p \ne 0, 2$ The prime field $\mathbb{F}_p$
Smallest size splitting field field:F3, i.e., the field with three elements.
Orbits over a splitting field under action of automorphism group orbit sizes: 1 (degree 1 representation), 1 (degree 1 representation), 2 (degree 1 representations), and 1 (degree 2 representation)
number: 4
Orbits over a splitting field under the multiplicative action of one-dimensional representations, i.e., up to projective equivalence orbit sizes: 4 (degree 1 representations), 1 (degree 2 representations)
number: 2
Other groups with the same character table quaternion group (see linear representation theory of quaternion group)

## Family contexts

Family name Parameter values General discussion of linear representation theory of family
dihedral group degree $n = 4$, order $2n = 8$ linear representation theory of dihedral groups
unitriangular matrix group of degree three over a finite field Case field:F2 linear representation theory of unitriangular matrix group of degree three over a finite field
COMPARE AND CONTRAST: View linear representation theory of groups of order 8 to compare and contrast the linear representation theory with other groups of order 8.

## Irreducible representations

### Summary information

Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2 , except in the last column, where we consider what happens in characteristic 2.

Name of representation type Number of representations of this type Degree Schur index Criterion for field Kernel (the normal subgroup of dihedral group:D8 that gets mapped to identity matrices -- see subgroup structure of dihedral group:D8) Quotient by kernel (on which it descends to a faithful representation) Characteristic 2
trivial 1 1 1 any whole group trivial group works
sign representation with kernel cyclic of order four 1 1 1 any cyclic maximal subgroup of dihedral group:D8: $\langle a \rangle$ cyclic group:Z2 works, same as trivial
sign representation with kernel a Klein four-subgroup 2 1 1 any Klein four-subgroups of dihedral group:D8: $\langle a^2, x \rangle$ or $\langle a^2, ax \rangle$ cyclic group:Z2 works, same as trivial
two-dimensional irreducible 1 2 1 any trivial subgroup, i.e., it is a faithful linear representation dihedral group:D8 indecomposable but not irreducible

### Trivial representation

The table below describes a one-dimensional (i.e., degree one) representation of the group. The matrices for the representation are uniquely determined, i.e., any equivalent representation must look exactly the same. The characteristic polynomial and minimal polynomial are also uniquely determined, i.e., they must be the same for any equivalent representation.

The trivial or principal representation is a one-dimensional representation sending every element of the group to the identity matrix of order one. This representation makes sense over all fields, and its character is 1 on all elements:

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value
$e$ $( 1 )$ $t - 1$ $t - 1$ 1
$a$ $( 1 )$ $t - 1$ $t - 1$ 1
$a^2$ $( 1 )$ $t - 1$ $t - 1$ 1
$a^3$ $( 1 )$ $t - 1$ $t - 1$ 1
$x$ $( 1 )$ $t - 1$ $t - 1$ 1
$ax$ $( 1 )$ $t - 1$ $t - 1$ 1
$a^2x$ $( 1 )$ $t - 1$ $t - 1$ 1
$a^3x$ $( 1 )$ $t - 1$ $t - 1$ 1

### Sign representations with kernels as the maximal normal subgroups

The table below describes a one-dimensional (i.e., degree one) representation of the group. The matrices for the representation are uniquely determined, i.e., any equivalent representation must look exactly the same. The characteristic polynomial and minimal polynomial are also uniquely determined, i.e., they must be the same for any equivalent representation.

The dihedral group has three normal subgroups of index two: the subgroup $\langle a \rangle$, the subgroup $\langle a^2, x \rangle$, and the subgroup $\langle a^2, ax \rangle$. For each such subgroup, there is an irreducible one-dimensional representation sending elements in that subgroup to $1$ and elements outside that subgroup to $-1$.

These representations make sense over all fields, but in characteristic two, they become the same as the trivial representation.

Here is the representation with kernel $\langle a \rangle$:

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value
$e$ $( 1 )$ $t - 1$ $t - 1$ 1
$a$ $( 1 )$ $t - 1$ $t - 1$ 1
$a^2$ $( 1 )$ $t - 1$ $t - 1$ 1
$a^3$ $( 1 )$ $t - 1$ $t - 1$ 1
$x$ $( -1 )$ $t + 1$ $t + 1$ -1
$ax$ $( -1 )$ $t + 1$ $t + 1$ -1
$a^2x$ $( -1 )$ $t + 1$ $t + 1$ -1
$a^3x$ $( -1 )$ $t + 1$ $t + 1$ -1

Here is the representation with kernel $\langle a^2, x \rangle$:

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value
$e$ $( 1 )$ $t - 1$ $t - 1$ 1
$a$ $( -1 )$ $t + 1$ $t + 1$ -1
$a^2$ $( 1 )$ $t - 1$ $t - 1$ 1
$a^3$ $( -1 )$ $t + 1$ $t + 1$ -1
$x$ $( 1 )$ $t - 1$ $t - 1$ 1
$ax$ $( -1 )$ $t + 1$ $t + 1$ -1
$a^2x$ $( 1 )$ $t - 1$ $t - 1$ 1
$a^3x$ $( -1 )$ $t + 1$ $t + 1$ -1

Here is the representation with kernel $\langle a^2, ax \rangle$:

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value
$e$ $( 1 )$ $t - 1$ $t - 1$ 1
$a$ $( -1 )$ $t + 1$ $t + 1$ -1
$a^2$ $( 1 )$ $t - 1$ $t - 1$ 1
$a^3$ $( -1 )$ $t + 1$ $t + 1$ -1
$x$ $( -1 )$ $t + 1$ $t + 1$ -1
$ax$ $( 1 )$ $t - 1$ $t - 1$ 1
$a^2x$ $( -1 )$ $t + 1$ $t + 1$ -1
$a^3x$ $( 1 )$ $t - 1$ $t - 1$ 1

### Two-dimensional irreducible representation

Further information: faithful irreducible representation of dihedral group:D8

The table below describes an irreducible representation of the group of degree more than one. The matrices for the representation are not uniquely determined -- we can choose alternative matrix descriptions by conjugating all matrices by a common matrix. The characteristic polynomial, minimal polynomial, trace (character), determinant, and eigenvalues for the matrices are, however, uniquely determined, since these are invariant under matrix conjugation.

The dihedral group of order eight has a two-dimensional irreducible representation, where the element $a$ acts as a rotation (by an angle of $\pi/2$), and the element $x$ acts as a reflection about the first axis. The matrices are:

$a \mapsto \begin{pmatrix}0 & -1 \\ 1 & 0 \\\end{pmatrix}, \qquad x \mapsto \begin{pmatrix}1 & 0 \\ 0 & -1 \\\end{pmatrix}.$

This particular choice of matrices give a representation as orthogonal matrices, and in fact, the representation is as signed permutation matrices (i.e., it takes values in the signed symmetric group of degree two). Thus, it is also a monomial representation.

Below is a description of the matrices based on the above choice as well as another formulation involving complex unitary matrices:

Element Matrix (orthogonal/monomial/signed permutation matrices) Matrix as complex unitary Characteristic polynomial Minimal polynomial Trace, character value Determinant
$e$ $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ $(t - 1)^2 = t^2 - 2t + 1$ $t - 1$ 2 1
$a$ $\begin{pmatrix}0 & -1 \\ 1 & 0 \\\end{pmatrix}$ $\begin{pmatrix}i & 0 \\ 0 & -i \\\end{pmatrix}$ $t^2 + 1$ $t^2 + 1$ 0 1
$a^2$ $\begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix}$ $\begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix}$ $(t + 1)^2 = t^2 + 2t + 1$ $t + 1$ -2 1
$a^3$ $\begin{pmatrix}0 & 1 \\ -1 & 0 \\\end{pmatrix}$ $\begin{pmatrix}-i & 0 \\ 0 & i \\\end{pmatrix}$ $t^2 + 1$ $t^2 + 1$ 0 1
$x$ $\begin{pmatrix}1 & 0 \\ 0 & -1 \\\end{pmatrix}$ $\begin{pmatrix} 0 & 1 \\ 1 & 0 \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0 -1
$ax$ $\begin{pmatrix}0 & 1 \\ 1 & 0 \\\end{pmatrix}$ $\begin{pmatrix} 0 & i \\ -i & 0 \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0 -1
$a^2x$ $\begin{pmatrix}-1 & 0 \\ 0 & 1 \\\end{pmatrix}$ $\begin{pmatrix} 0 & -1 \\ -1 & 0 \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0 -1
$a^3x$ $\begin{pmatrix}0 & -1 \\ -1 & 0 \\\end{pmatrix}$ $\begin{pmatrix} 0 & -i \\ i & 0 \\\end{pmatrix}$ $t^2 - 1$ $t^2 - 1$ 0 -1
Set of values used $\{ 0,1,-1 \}$ $\{ 0,1,-1,i,-i \}$ -- -- $\{ 2,-2,0 \}$ $\{ 1,-1 \}$
Ring generated by values used (characteristic zero) $\mathbb{Z}$ -- ring of integers $\mathbb{Z}[i]$ -- ring of Gaussian integers -- -- $\mathbb{Z}$ -- ring of integers $\mathbb{Z}$ -- ring of integers
Field generated by values used (characteristic zero) $\mathbb{Q}$ -- field of rational numbers $\mathbb{Q}(i) = \mathbb{Q}[t]/(t^2 + 1)$ -- -- $\mathbb{Q}$ -- field of rational numbers $\mathbb{Q}$ -- field of rational numbers

## Character table

FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):
Orthogonality relations: Character orthogonality theorem | Column orthogonality theorem
Separation results (basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zero
Numerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integers
Character value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma

This character table works over characteristic zero:

Representation/Conj class $\{e \}$ (size 1) $\{ a^2 \}$ (size 1) $\{ a, a^{-1} \}$ (size 2) $\{ x, a^2x \}$ (size 2) $\{ ax, a^3x \}$ (size 2)
Trivial representation 1 1 1 1 1
$\langle a \rangle$-kernel 1 1 1 -1 -1
$\langle a^2, x \rangle$-kernel 1 1 -1 1 -1
$\langle a^2, ax\rangle$-kernel 1 1 -1 -1 1
2-dimensional 2 -2 0 0 0

The same character table works over any characteristic not equal to 2 where the elements 1,-1,0,2,-2 are interpreted over the field.

Here is the size-degree-weighted character table, i.e., each cell entry is obtained by multiplying the character value by the size of the conjugacy class and then dividing by the degree of the representation. Note that size-degree-weighted characters are algebraic integers.

Representation/Conj class $\{ e \}$ (size 1) $\{ a^2\}$ (size 1) $\{ a, a^{-1} \}$ (size 2) $\{ x, a^2x \}$ (size 2) $\{ ax, a^3x \}$ (size 2)
Trivial representation 1 1 2 2 2
$\langle a \rangle$-kernel 1 1 2 -2 -2
$\langle a^2, x \rangle$-kernel 1 1 -2 2 -2
$\langle a^2, ax\rangle$-kernel 1 1 -2 -2 2
2-dimensional 1 -1 0 0 0

## Table of matrix entries

This table satisfies the grand orthogonality theorem. Note that unlike the character table, this table is not canonical but rather, for the degree two irreducible representation, depends on the choice of basis.

Representation/element $e$ $a$ $a^2$ $a^3$ $x$ $ax$ $a^2x$ $a^3x$
trivial 1 1 1 1 1 1 1 1
sign with kernel $\langle a \rangle$ 1 1 1 1 -1 -1 -1 -1
sign with kernel $\langle a^2,x \rangle$ 1 -1 1 -1 1 -1 1 -1
sign with kernel $\langle a^2, ax \rangle$ 1 -1 1 -1 -1 1 -1 1
faithful irreducible representation of degree two -- top left entry 1 0 -1 0 1 0 -1 0
faithful irreducible representation of degree two -- top right entry 0 -1 0 1 0 1 0 -1
faithful irreducible representation of degree two -- bottom left entry 0 1 0 -1 0 1 0 -1
faithful irreducible representation of degree two -- bottom right entry 1 0 -1 0 -1 0 1 0

## Realizability information

TERMINOLOGY AND FACTS TO CHECK AGAINST:
Terminology: ring generated by character values | minimal ring of realization of irreducible representations
Facts: linear representation is realizable over principal ideal domain iff it is realizable over field of fractions

### Smallest ring of realization

Representation Smallest ring over which it is realized Smallest set of elements in matrix entries
trivial representation $\mathbb{Z}$ -- ring of integers $\{ 1 \}$
$\langle a \rangle$-kernel $\mathbb{Z}$ -- ring of integers $\{ 1, -1 \}$
$\langle a^2,x \rangle$-kernel $\mathbb{Z}$ -- ring of integers $\{ 1, -1 \}$
$\langle a^2,ax \rangle$-kernel $\mathbb{Z}$ -- ring of integers $\{ 1,-1 \}$
two-dimensional irreducible $\mathbb{Z}$ -- ring of integers $\{ 1,0,-1 \}$

### Smallest ring of realization as orthogonal matrices

Representation Smallest ring over which it is realized Smallest set of elements in matrix entries
trivial representation $\mathbb{Z}$ -- ring of integers $\{ 1 \}$
$\langle a \rangle$-kernel $\mathbb{Z}$ -- ring of integers $\{ 1, -1 \}$
$\langle a^2,x \rangle$-kernel $\mathbb{Z}$ -- ring of integers $\{ 1,-1 \}$
$\langle a^2,ax \rangle$-kernel $\mathbb{Z}$ -- ring of integers $\{ 1,-1 \}$
two-dimensional irreducible $\mathbb{Z}$ -- ring of integers $\{ 1,0,-1 \}$

## Orthogonality relations and numerical checks

General statement Verification in this case
number of irreducible representations equals number of conjugacy classes Both numbers are equal to 5
sufficiently large implies splitting: if the field has characteristic not dividing the order of the group and has primitive $d^{th}$ roots of unity for $d$ the exponent of the group, it is a splitting field. In fact, for this group, any field of characteristic not 2 is a splitting field.
number of one-dimensional representations equals order of abelianization The number of one-dimensional representations equals $4$, which is the order of the abelianization, which is the quotient by center of dihedral group:D8 and is a Klein four-group
sum of squares of degrees of irreducible representations equals group order $1^2 + 1^2 + 1^2 + 1^2 + 2^2 = 8$.
degree of irreducible representation divides order of group The degrees (1,1,1,1,2) all divide the order 8.
degree of irreducible representation divides index of abelian normal subgroup The degrees (1,1,1,1,2) all divide the index 2 of the abelian normal subgroups: cyclic maximal subgroup of dihedral group:D8 and Klein four-subgroups of dihedral group:D8.
order of inner automorphism group bounds square of degree of irreducible representation The degree are all at most 2, the square of which is 4. The inner automorphism group is the quotient by center of dihedral group:D8, and is a Klein four-group of order 4.
row orthogonality theorem and the column orthogonality theorem can be verified from the character table.

## Action of automorphism group

The automorphism group of the dihedral group preserves the trivial representation, the two-dimensional representation, and the sign representation whose kernel is the cyclic group $\langle a \rangle$. The two sign representations with kernels $\langle a^2, x \rangle$ and $\langle a^2,ax\rangle$ are exchanged by an outer automorphism.

## Isoclinism and projective representations

Please compare this with projective representation theory of Klein four-group.

### Grouping by restriction to center

Restriction to center as a representation of center of dihedral group:D8 which is isomorphic to cyclic group:Z2 (this determines, essentially, the cohomology class of the projective representation for the inner automorphism group) List of irreducible projective representations of the inner automorphism group (which is Klein four-group) List of corresponding linear representations of dihedral group:D8 List of degrees Sum of squares of degrees (should equal order of inner automorphism group, which is 4)
trivial representation trivial representation trivial representation, and the three one-dimensional representations with kernels the subgroups of order four 1,1,1,1 4
sign representation nontrivial two-dimensional projective representation faithful irreducible representation of dihedral group:D8 2 4

### Grouping by projective representation

Irreducible projective representation of inner automorphism group (which is Klein four-group) Degree Size of stabilizer under action of one-dimensional representations of dihedral group:D8 Size of orbit (equals order of abelianization divided by size of stabilizer) List of irreducible representations of dihedral group:D8
trivial representation 1 1 4 trivial representation, and the three one-dimensional representations with kernels the subgroups of order four
nontrivial two-dimensional representation 2 4 1 faithful irreducible representation of dihedral group:D8

## Group ring interpretation

### Direct sum decomposition

If $K$ is any field whose characteristic is not 2, then the group ring $K[D_8]$ splits as a direct sum of two-sided ideals corresponding to the irreducible representations:

$K[D_8] \cong M_1(K) \oplus M_1(K) \oplus M_1(K) \oplus M_1(K) \oplus M_2(K) = K \oplus K \oplus K \oplus K \oplus M_2(K)$

More generally, if $R$ is any commutative unital ring that is uniquely 2-divisible, then we can write:

$R[D_8] \cong M_1(R) \oplus M_1(R) \oplus M_1(R) \oplus M_1(R) \oplus M_2(R) = R \oplus R \oplus R \oplus R \oplus M_2(R)$

Note that the ring of integers $\mathbb{Z}$ does not satisfy the condition for this direct sum decomposition to hold. Instead we need to use the ring $\mathbb{Z}[1/2]$ (In general, we need to use a ring that is uniquely divisible by all primes dividing the order of the group).

### Explicit decomposition and idempotents

We can write:

$R[D_8] = M_1(R)e_1 \oplus M_1(R)e_2 \oplus M_1(R)e_3 \oplus M_1(R)e_4 \oplus M_2(R)e_5$

where $e_1,e_2,e_3,e_4,e_5$ are idempotents. These are called primitive central idempotents.

Note that $e$ here denotes the identity of the group, and can also be written as $1$ since it gives the identity of the group ring.

Representation Degree $d$ Corresponding primitive central idempotent that gives the identity element in the corresponding direct summand $M_d(R)$ How to read this from the character table
trivial representation 1 $\frac{e + a + a^2 + a^3 + x + ax + a^2x + a^3x}{8} = \frac{(1 + a + a^2 + a^3)(1 + x)}{8}$ We multiply each group element by its character value, add up, and divide by the order of the group. For the trivial representation, the character values are all 1.
sign representation with kernel $\langle a \rangle$ 1 $\frac{e + a + a^2 + a^3 - x - ax - a^2x - a^3x}{8}$, which can be written as $\frac{(1 + a + a^2 + a^3)(1 - x)}{8}$ For this representation, the character values are 1 on $\langle a \rangle$ and -1 outside.
sign representation with kernel $\langle a^2,x \rangle$ 1 $\frac{e - a + a^2 - a^3 + x - ax + a^2x - a^3x}{8} = \frac{(1 - a)(1 + a^2)(1 + x)}{8}$ For this representation, the character values are 1 on $\langle a^2,x \rangle$ and -1 outside.
sign representation with kernel $\langle a^2, ax \rangle$ 1 $\frac{e - a + a^2 - a^3 - x + ax - a^2x + a^3x}{8}$ For this representation, the character values are 1 on $\langle a^2,ax \rangle$ and -1 outside.
faithful irreducible representation of dihedral group:D8 2 $\frac{2e - 2a^2}{8}$, which can also be written as $\frac{1 - a^2}{4}$ The character value is 2 on the identity, -2 on $a^2$, and 0 elsewhere.

## Relation with representations of subgroups

### Induced representations from subgroups

Since the dihedral group is a finite nilpotent group, it is in particular a finite supersolvable group, and hence, it is a monomial-representation group: every irreducible representation can be realized as a monomial representation, i.e., every irreducible representation is induced from a degree one representation of a subgroup. (Point (5) below explains how the two-dimensional irreducible representation is induced).

1. The trivial representation on the center induces a representation obtained as a sum of the four one-dimensional representations.
2. The sign representation on the center (which comprises $\pm 1$) induces the double of the two-dimensional irreducible representation of the dihedral group.
3. The trivial representation on the cyclic subgroup generated by $a$ induces a representation on the whole group that is the sum of a trivial representation and the representation with the $a$-kernel.
4. A representation on $\langle a \rangle$ that sends $a$ to $-1$ induces a representation of the whole group that is the sum of the sign representations for the other two kernels.
5. A representation on $\langle a \rangle$ that sends $a$ to $i$ (now viewed as a complex number) induces the two-dimensional irreducible representation.

### Verification of Artin's induction theorem

Artin's induction theorem states that the characters induced from characters on cyclic subgroups span the space of class functions. Points (2) and (5) cover the case of the two-dimensional irreducible representation. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

## GAP implementation

### Degrees of irreducible representations

These can be computed using the CharacterDegrees function:

gap> CharacterDegrees(DihedralGroup(8));
[ [ 1, 4 ], [ 2, 1 ] ]

### Character table

The characters of the irreducible representations can be computed using Irr and CharacterTable:

gap> Irr(CharacterTable(DihedralGroup(8)));
[ Character( CharacterTable( <pc group of size 8 with 3 generators> ),
[ 1, 1, 1, 1, 1 ] ), Character( CharacterTable( <pc group of size 8 with
3 generators> ), [ 1, -1, 1, 1, -1 ] ),
Character( CharacterTable( <pc group of size 8 with 3 generators> ),
[ 1, 1, -1, 1, -1 ] ), Character( CharacterTable( <pc group of size
8 with 3 generators> ), [ 1, -1, -1, 1, 1 ] ),
Character( CharacterTable( <pc group of size 8 with 3 generators> ),
[ 2, 0, 0, -2, 0 ] ) ]

The character table can be displayed more nicely using the Display function:

gap> Display(CharacterTable(DihedralGroup(8)));
CT1

2  3  2  2  3  2

1a 2a 4a 2b 2c

X.1     1  1  1  1  1
X.2     1 -1  1  1 -1
X.3     1  1 -1  1 -1
X.4     1 -1 -1  1  1
X.5     2  .  . -2  .

### Irreducible representations

The irreducible representations can be accessed using GAP's IrreducibleRepresentations function:

gap> IrreducibleRepresentations(DihedralGroup(8));
[ Pcgs([ f1, f2, f3 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
Pcgs([ f1, f2, f3 ]) -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
Pcgs([ f1, f2, f3 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ] ],
Pcgs([ f1, f2, f3 ]) -> [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ] ],
Pcgs([ f1, f2, f3 ]) ->
[ [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ],
[ [ -1, 0 ], [ 0, -1 ] ] ] ]