Linear representation theory of M16

This article gives specific information, namely, linear representation theory, about a particular group, namely: M16.
View linear representation theory of particular groups | View other specific information about M16

This article discusses the linear representation theory of the group M16, a group of order 16 given by the presentation:

$G = M_{16} := \langle a,x \mid a^8 = x^2 = e, xax^{-1} = a^5 \rangle$

Summary

Item	Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$)	1,1,1,1,1,1,1,1,2,2 (1 occurs 8 times, 2 occurs 2 times) maximum: 2, lcm: 2, number: 10, sum of squares: 16
Schur index values of irreducible representations	1 (all of them)
smallest ring of realization (characteristic zero)	$\mathbb{Z}[i] = \mathbb{Z}[\sqrt{-1}] = \mathbb{Z}[t]/(t^2 + 1)$ -- ring of Gaussian integers
ring generated by character values (characteristic zero)	$\mathbb{Z}[2i] = \mathbb{Z}[\sqrt{-4}] = \mathbb{Z}[t]/(t^2 + 4)$
minimal splitting field, i.e., smallest field of realization (characteristic zero)	$\mathbb{Q}(i) = \mathbb{Q}(\sqrt{-1}) = \mathbb{Q}[t]/(t^2 + 1)$ Same as field generated by character values, because all Schur index values are 1.
condition for a field to be a splitting field	The characteristic should not be equal to 2, and the polynomial $t^2 + 1$ should split. For a finite field of size $q$, this is equivalent to saying that $q \equiv 1 \pmod 4$
minimal splitting field in characteristic $p \ne 0,2$	Case $p \equiv 1 \pmod 4$: prime field $\mathbb{F}_p$ Case $p \equiv 3 \pmod 4$: Field $\mathbb{F}_{p^2}$, quadratic extension of prime field
smallest size splitting field	Field:F5, i.e., the field with five elements.
degrees of irreducible representations over the rational numbers	1,1,1,1,2,2,4 (1 occurs 4 times, 2 occurs 2 times, 4 occurs 1 time) number: 7
orbits of irreducible representations over a splitting field under action of automorphism group	2 orbits of size 1 of degree 1 representations, 3 orbits of size 2 of degree 1 representations, 1 orbit of size 2 of degree 2 representations. number: 6

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 (a normal subgroup of M16 comprising the elements that map to identity matrices; see subgroup structure of M16)	Quotient by kernel (on which it descends to a faithful representation)	Characteristic 2?
trivial	1	1	1	any	whole group	trivial group	works
sign, kernel a non-cyclic maximal subgroup	1	1	1	any	direct product of Z4 and Z2 in M16 -- $\langle a^2, x \rangle$	cyclic group:Z2	works, same as trivial
sign, kernel a cyclic maximal subgroup	2	1	1	any	Z8 in M16 -- either $\langle a \rangle$ or $\langle ax \rangle$	cyclic group:Z2	works, same as trivial
representation with kernel $\langle a^2x \rangle$	2	1	1	must contain a primitive fourth root of unity, or equivalently, $t^2 + 1$ must split	non-central Z4 in M16	cyclic group:Z4	works, same as trivial
representation with kernel $\langle a^4, x \rangle$	2	1	1	must contain a primitive fourth root of unity, or equivalently, $t^2 + 1$ must split	V4 in M16	cyclic group:Z4	works, same as trivial
faithful irreducible representation of M16	2	2	1	must contain a primitive fourth root of unity, or equivalently, $t^2 + 1$ must split	trivial subgroup	M16	indecomposable but not irreducible

Below are representations that are irreducible over a non-splitting field, but split over a splitting field.

Name of representation type	Number of representations of this type	Degree	Criterion for field	What happens over a splitting field?	Kernel	Quotient by kernel (on which it descends to a faithful representation)
two-dimensional representation with kernel $\langle a^2x \rangle$	1	2	must not contain a primitive fourth root of unity, or equivalently, $t^2 + 1$ does not split	splits into the one-dimensional representations with kernel $\langle a^2x \rangle$	non-central Z4 in M16	cyclic group:Z4
two-dimensional representation with kernel $\langle a^4,x \rangle$	1	2	must not contain a primitive fourth root of unity, or equivalently, $t^2 + 1$ does not split	splits into the one-dimensional representations with kernel $\langle a^4,x \rangle$	Klein four-group	cyclic group:Z4
four-dimensional representation	1	4	must not contain a primitive fourth root of unity, or equivalently, $t^2 + 1$ does not split	splits into the two faithful irreducible representations of degree two	trivial subgroup	M16

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

Below is the character table over a splitting field. Here $i$ denotes a square root of $-1$ in the field.

Representation/conjugacy class and size	$\{ e \}$ (size 1)	$\{ a^4 \}$ (size 1)	$\{ a^2 \}$ (size 1)	$\{ a^6 \}$ (size 1)	$\{ a, a^5 \}$ (size 2)	$\{ a^3, a^7 \}$ (size 2)	$\{ ax, a^5x \}$ (size 2)	$\{ a^3x, a^7x \}$ (size 2)	$\{ x, a^4x\}$ (size 2)	$\{ a^2x, a^6x \}$ (size 2)
trivial	1	1	1	1	1	1	1	1	1	1
$\langle a^2,x \rangle$-kernel	1	1	1	1	-1	-1	-1	-1	1	1
$\langle a \rangle$-kernel	1	1	1	1	1	1	-1	-1	-1	-1
$\langle ax \rangle$-kernel	1	1	1	1	-1	-1	1	1	-1	-1
$\langle a^2x \rangle$-kernel (first)	1	1	-1	-1	$i$	$-i$	$-i$	$i$	-1	1
$\langle a^2x \rangle$-kernel (second)	1	1	-1	-1	$-i$	$i$	$i$	$-i$	-1	1
$\langle a^4,x \rangle$-kernel (first)	1	1	-1	-1	$i$	$-i$	$i$	$-i$	1	-1
$\langle a^4,x \rangle$-kernel (second)	1	1	-1	-1	$-i$	$i$	$-i$	$i$	1	-1
faithful irreducible representation of M16 (first)	2	-2	$2i$	$-2i$	0	0	0	0	0	0
faithful irreducible representation of M16 (second)	2	-2	$-2i$	$2i$	0	0	0	0	0	0

Below are the size-degree-weighted characters, obtained by multiplying each character value by the size of the conjugacy class and dividing by the degree of the irreducible representation:

Representation/conjugacy class and size	$\{ e \}$ (size 1)	$\{ a^4 \}$ (size 1)	$\{ a^2 \}$ (size 1)	$\{ a^6 \}$ (size 1)	$\{ a, a^5 \}$ (size 2)	$\{ a^3, a^7 \}$ (size 2)	$\{ ax, a^5x \}$ (size 2)	$\{ a^3x, a^7x \}$ (size 2)	$\{ x, a^4x\}$ (size 2)	$\{ a^2x, a^6x \}$ (size 2)
trivial	1	1	1	1	2	2	2	2	2	2
$\langle a \rangle$-kernel	1	1	1	1	2	2	-2	-2	-2	-2
$\langle ax \rangle$-kernel	1	1	1	1	-2	-2	2	2	-2	-2
$\langle a^2,x \rangle$-kernel	1	1	1	1	-2	-2	-2	-2	2	2
$\langle a^2x \rangle$-kernel (first)	1	1	-1	-1	$2i$	$-2i$	$-2i$	$2i$	-2	2
$\langle a^2x \rangle$-kernel (second)	1	1	-1	-1	$-2i$	$2i$	$2i$	$-2i$	-2	2
$\langle a^4,x \rangle$-kernel (first)	1	1	-1	-1	$2i$	$-2i$	$2i$	$-2i$	2	-2
$\langle a^4,x \rangle$-kernel (second)	1	1	-1	-1	$-2i$	$2i$	$-2i$	$2i$	2	-2
faithful irreducible representation of M16	1	-1	$i$	$-i$	0	0	0	0	0	0
faithful irreducible representation of M16 (second)	1	-1	$-i$	$i$	0	0	0	0	0	0

GAP implementation

Degrees of irreducible representations

The degrees of irreducible representation can be computed using the CharacterDegrees function:

gap> CharacterDegrees(SmallGroup(16,6));
[ [ 1, 8 ], [ 2, 2 ] ]

Character table

The character table can be computed using the Irr and CharacterTable functions:

gap> Irr(CharacterTable(SmallGroup(16,6)));
[ Character( CharacterTable( <pc group of size 16 with 4 generators> ),
    [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
  Character( CharacterTable( <pc group of size 16 with 4 generators> ),
    [ 1, -1, 1, 1, 1, -1, -1, 1, 1, -1 ] ),
  Character( CharacterTable( <pc group of size 16 with 4 generators> ),
    [ 1, 1, -1, 1, 1, -1, 1, -1, 1, -1 ] ),
  Character( CharacterTable( <pc group of size 16 with 4 generators> ),
    [ 1, -1, -1, 1, 1, 1, -1, -1, 1, 1 ] ),
  Character( CharacterTable( <pc group of size 16 with 4 generators> ),
    [ 1, E(4), 1, -1, 1, E(4), -E(4), -1, -1, -E(4) ] ),
  Character( CharacterTable( <pc group of size 16 with 4 generators> ),
    [ 1, -E(4), 1, -1, 1, -E(4), E(4), -1, -1, E(4) ] ),
  Character( CharacterTable( <pc group of size 16 with 4 generators> ),
    [ 1, E(4), -1, -1, 1, -E(4), -E(4), 1, -1, E(4) ] ),
  Character( CharacterTable( <pc group of size 16 with 4 generators> ),
    [ 1, -E(4), -1, -1, 1, E(4), E(4), 1, -1, -E(4) ] ),
  Character( CharacterTable( <pc group of size 16 with 4 generators> ),
    [ 2, 0, 0, 2*E(4), -2, 0, 0, 0, -2*E(4), 0 ] ),
  Character( CharacterTable( <pc group of size 16 with 4 generators> ),
    [ 2, 0, 0, -2*E(4), -2, 0, 0, 0, 2*E(4), 0 ] ) ]

A nicer display can be achieved using the Display function:

gap> Display(CharacterTable(SmallGroup(16,6)));
CT3

      2  4  3  3  4  4  3  3  3  4  3

        1a 8a 2a 4a 2b 8b 8c 4b 4c 8d

X.1      1  1  1  1  1  1  1  1  1  1
X.2      1 -1  1  1  1 -1 -1  1  1 -1
X.3      1  1 -1  1  1 -1  1 -1  1 -1
X.4      1 -1 -1  1  1  1 -1 -1  1  1
X.5      1  A  1 -1  1  A -A -1 -1 -A
X.6      1 -A  1 -1  1 -A  A -1 -1  A
X.7      1  A -1 -1  1 -A -A  1 -1  A
X.8      1 -A -1 -1  1  A  A  1 -1 -A
X.9      2  .  .  B -2  .  .  . -B  .
X.10     2  .  . -B -2  .  .  .  B  .

A = E(4)
  = ER(-1) = i
B = 2*E(4)
  = 2*ER(-1) = 2i

Irreducible representations

The irreducible representations can be computed explicitly using the IrreducibleRepresentations function:

gap> IrreducibleRepresentations(SmallGroup(16,6));
[ Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
  Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
  Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
  Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ],
  Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ E(4) ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ]
     ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -E(4) ] ], [ [ 1 ] ], [ [ -1 ] ],
      [ [ 1 ] ] ],
  Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ E(4) ] ], [ [ -1 ] ], [ [ -1 ] ],
      [ [ 1 ] ] ],
  Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -E(4) ] ], [ [ -1 ] ], [ [ -1 ] ],
      [ [ 1 ] ] ],
  Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, E(4) ], [ 1, 0 ] ], [ [ 1, 0 ],
          [ 0, -1 ] ], [ [ E(4), 0 ], [ 0, E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ]
    ,
  Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, -E(4) ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0,
              -1 ] ], [ [ -E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ]
     ] ]