Linear representation theory of quaternion group

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

Summary

The quaternion group is one of the few examples of a rational group that is not a rational-representation group. In other words, all its characters over the complex numbers are rational-valued, but not every representation of it can be realized over the rationals.

The character table of the quaternion group is the same as that of the dihedral group of order eight. Note, however, that the fields of realization for the representations differ, because one of the representations of the quaternion group has Schur index two.

Item Value
Degrees of irreducible representations over a splitting field (such as or ) 1,1,1,1,2
maximum: 2, lcm: 2, number: 5, sum of squares: 8
Schur index values of irreducible representations 1,1,1,1,2 (characteristic zero)
maximum: 2, lcm: 2
1,1,1,1,1 (characteristic other than 0,2)
Smallest ring of realization for all irreducible representations (characteristic zero) There are multiple candidates. where is a square root of , equivalently , the ring of Gaussian integers is one candidate. Another is or .
More generally, any ring of the form where is a ring of realization for all irreducible representations. In particular, works for any rational .
Minimal splitting field (i.e., field of realization) for all irreducible representations (characteristic zero) There are multiple candidates. or works, so does or . More generally, where is a splitting field. In particular, works for any rational .
See minimal splitting field need not be unique, minimal splitting field need not be cyclotomic
Ring generated by character values (characteristic zero)
Field generated by character values (characteristic zero) (hence it is a rational group)
See also: Field generated by character values need not be a splitting field|rational not implies rational-representation
Condition for being a splitting field for this group Sufficient condition: the characteristic is not two and there exist in the field such that . In particular, every finite field of characteristic not two is a splitting field, because every element of a finite field is expressible as a sum of two squares and in particular, is a sum of two squares in any finite field.
Minimal splitting field (characteristic ) The prime field
Smallest size splitting field field:F3, i.e., the field of three elements.
Orbit structure of irreducible representations over splitting field under automorphism group orbit sizes: 1 (degree 1 representation), 3 (degree 1 representations), 1 (degree 2 representation)
number: 3
Orbit structure of irreducible representations over splitting field under multiplicative action of one-dimensional representations, i.e., up to projective equivalence orbit sizes: 4 (degree 1 representations), 1 (degree 2 representation)
number: 2
Degrees of irreducible representations over a non-splitting field, e.g., the field of rational numbers or the field of real numbers 1,1,1,1,4
number: 5
Groups with same character table Dihedral group:D8


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 quaternion group that gets mapped to identity matrices -- see subgroup structure of quaternion group) Quotient by kernel (on which it descends to a faithful representation). Must be a linearly primitive group. Characteristic 2
trivial 1 1 1 any whole group trivial group works
sign representation with kernel cyclic of order four 3 1 1 any cyclic maximal subgroups of quaternion group: , , cyclic group:Z2 works, same as trivial
two-dimensional irreducible 1 2 2 being a sum of two squares is a sufficient condition. Because every element of a finite field is expressible as a sum of two squares, all finite fields work. Also, formally real fields do not work trivial subgroup, i.e., it is a faithful linear representation quaternion group  ?

Below are representations that are irreducible over a non-splitting field, but split over a splitting field. Again, we assume that the characteristic is not 2.

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)
four-dimensional faithful irreducible 1 4 Not a splitting field splits into two copies of the two-dimensional faithful irreducible. trivial subgroup, i.e., it is a faithful linear representation quaternion group

Trivial representation

The trivial or principal representation is a one-dimensional representation that sends every element of the group to the 1-by-1 matrix 1. This representation makes sense over any field and in fact over any unital ring.

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value
1
1
1
1
1
1
1
1


Sign representations with -kernels

The quaternion group has three maximal normal subgroups: the cyclic subgroups generated by respectively. For each maximal normal subgroup, we obtain a one-dimensional representation with that subgroup as kernel. The representation sends elements inside the subgroup to 1, and elements outside the subgroup to -1.

The representations are detailed below:

Representation with -kernel:

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value, determinant (all same because it is degree 1)
1
1
1
1
-1
-1
-1
-1

Representation with -kernel:

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value, determinant (all same because it is degree 1)
1
1
-1
-1
1
1
-1
-1

Representation with -kernel:

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value, determinant (all same because it is degree 1)
1
1
-1
-1
-1
-1
1
1

Two-dimensional irreducible representation over a splitting field

Further information: Faithful irreducible representation of quaternion group


Suppose are elements of a field (or more generally, a commutative unital ring) such that , then the representation can be realized in terms of the entries . The explicit representation involving the Hamiltonian quaternions is the special case of this where is a square root of and . Another case is , . A third case of interest is , which contains primitive cube roots of unity, where we set and as distinct primitive cube roots of unity.

Note that the representation makes sense in all characteristics, but there are some problems interpreting it in characteristic two.

Element Matrix using with Matrix using . Obtained by setting Matrix using
. Obtained by setting
Matrix using primitive cube roots of unity, which we denote by and , so , . Characteristic polynomial Minimal polynomial Trace, character value Determinant
2 1
-2 1
0 1
0 1
0 1
0 1
0 1
0 1
Set of values used -- --
Ring generated by values used (characteristic zero) or or -- --
Field generated by values used (characteristic zero) or or , same as or -- --


Four-dimensional irreducible representation over a non-splitting field

The quaternion group has no irreducible two-dimensional representation over the reals (see faithful irreducible representation of quaternion group#Frobenius-Schur indicator). However, it has a four-dimensional representation over the reals, which splits over the complex numbers as a direct sum of two copies of the two-dimensional irreducible representation over the complex numbers. This representation is obtained by viewing the Hamiltonian quaternions as a four-dimensional vector space over the real numbers, and writing the matrices for left multiplication by the elements of the quaternion group. The typical choice of basis is . Note that multiplication by anything other than gives a matrix with zeros on the diagonal, hence the character is zero on all elements outside the center.

Note that this representation is actually a representation over the rational numbers, and all its entries are signed permutation matrices, i.e., matrices with exactly one nonzero entry in every row and every column and that entry is .

Element Matrix Characteristic polynomial Minimal polynomial Trace, character value
4
-4
0
0
0
0
0
0

This representation is irreducible over any formally real field (is it? verify).

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 and over any other characteristic not equal to two once we reduce the entries mod the characteristic:

Representation/Conjugacy class (identity; size 1) (size 1) (size 2) (size 2) (size 2)
Trivial representation 1 1 1 1 1
-kernel 1 1 1 -1 -1
-kernel 1 1 -1 1 -1
-kernel 1 1 -1 -1 1
2-dimensional 2 -2 0 0 0


The size-degree-weighted characters are given as follows, where a size-degree-weighted character value is obtained by multiplying the character value by the size of the conjugacy class and dividing by the degree of the representation. Note that size-degree-weighted characters are algebraic integers:

Representation/Conjugacy class (identity; size 1) (size 1) (size 2) (size 2) (size 2)
trivial representation 1 1 2 2 2
-kernel 1 1 2 -2 -2
-kernel 1 1 -2 2 -2
-kernel 1 1 -2 -2 2
2-dimensional 1 -1 0 0 0

Degrees of irreducible representations

Below is a description of the degrees of irreducible representations over fields of characteristic not equal to .

Type of field Condition Condition on for finite field of size Condition in characteristic zero Degrees of irreducible representations
splitting field sufficient condition: -1 is a sum of two squares all finite fields of characteristic not equal to 2 (see every element of a finite field is expressible as a sum of two squares) sufficient condition: any field where -1 is a sum of two squares 1,1,1,1,2
not a splitting field sufficient condition: subfield of reals no finite field sufficient condition: subfield of reals (see faithful irreducible representation of quaternion group#Frobenius-Schur indicator to learn why one of the irreducible representations cannot be realized over the reals) 1,1,1,1,4

Realizability information

Smallest ring of realization

Representation Smallest ring of realization Smallest set of elements occurring as entries of matrices
trivial representation -- ring of integers
-kernel -- ring of integers
-kernel -- ring of integers
-kernel -- ring of integers
two-dimensional irreducible (over splitting field) ; ; where if over ; if over ; if over
four-dimensional irreducible (over non-splitting field) -- ring of integers

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 a field has primitive roots where is the exponent of the group, it is a splitting field. In this case, , and having a primitive fourth root is equivalent to having a square root, and a field that has this property is a splitting field. However, the condition is not necessary, i.e., there are splitting fields that are not sufficiently large, such as field:F3.
number of one-dimensional representations equals order of abelianization Both numbers are equal to 4. The derived subgroup is (see center of quaternion group) and the quotient group (i.e., the abelianization) is a Klein four-group, having order four.
sum of squares of degrees of irreducible representations equals order of group .
degree of irreducible representation divides index of abelian normal subgroup All degrees divide 2, which is the index of the abelian normal subgroup (and also of and .
order of inner automorphism group bounds square of degree of irreducible representation The center is (see center of quaternion group) and the quotient group (i.e., the inner automorphism group) (a Klein four-group) has order four. The degrees of irreducible representations, 1,1,1,1,2, all have the property that the square is at most 4.
row orthogonality theorem and the column orthogonality theorem can be verified from the character table.

Action of automorphisms and endomorphisms

FACTS TO CHECK AGAINST for action of automorphism group on irreducible representations and degrees of irreducible representations
number of orbits of irreducible representations equals number of orbits under automorphism group | number of orbits of irreducible representations equals number of orbits of conjugacy classes under any subgroup of automorphism group
cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group

Splitting field of characteristic not two

The automorphism group of the quaternion group permutes the three sign representations. In fact, this automorphism group permutes the sign representations in precisely the same way as it permutes the three maximal normal subgroups.

The trivial representation and the two-dimensional representation remain invariant under all automorphisms.

The orbit sizes are thus: one size 1 orbit of degree 1 representations (trivial representation), one size 3 orbit of degree 1 representations (sign representations), and one size 1 orbit of degree 2 representations (the faithful irreducible representation).

Non-splitting field of characteristic not two

The automorphism group permutes the three sign representation. The trivial representation and the four-dimensional irreducible representation remain invariant.

The orbit sizes are thus: one size 1 orbit of degree 1 representations (trivial representation), one size 3 orbit of degree 1 representations (sign representations), and one size 1 orbit of degree 4 representations (the faithful irreducible representation that becomes reducible over a splitting field).

Relation with quotients

The quaternion group has six normal subgroups: the whole group, the trivial subgroup, center of quaternion group, and three cyclic maximal subgroups of quaternion group. The irreducible representations with kernel a particular normal subgroup correspond precisely to the faithful irreducible representations of the quotient group; the irreducible representations with kernel containing a particular normal subgroup correspond precisely to the irreducible representations of the quotient group. Information in this regard is presented below:

Q8latticeofsubgroups.png

Normal subgroup in whole group Normal subgroup isomorphism type Quotient group Number of such normal subgroups Linear representation theory of quotient group Degrees of irreducible representations of quotient group List of irreducible representations of quotient group (corresponding representation of whole group) Degrees of faithful irreducible representations of quotient group List of faithful irreducible representations of quotient group (corresponding representation of whole group)
whole group quaternion group trivial group 1 1 trivial (trivial) 1 trivial (trivial)
cyclic maximal subgroups of quaternion group cyclic group:Z4 cyclic group:Z2 3 linear representation theory of cyclic group:Z2 1,1 trivial (trivial), sign (-kernel, -kernel, and -kernel sign depending on which subgroup is chosen) 1 sign (-kernel, -kernel, and -kernel sign depending on which subgroup is chosen)
center of quaternion group cyclic group:Z2 Klein four-group 1 linear representation theory of Klein four-group 1,1,1,1 trivial (trivial), three sign representations for the various copies of Z2 in V4 (correspond to the representations with , , and kernels) -- --
trivial subgroup trivial group quaternion group 1 current page 1,1,1,1,2 that's what this page is about 2 two-dimensional irreducible (two-dimensional irreducible)

Group ring interpretation

For a splitting field

Further information: Group ring over splitting field is direct sum of matrix rings for each irreducible representation

Over a splitting field (which could be a finite field or a field such as or ), we have the decomposition (as a direct product of rings, or equivalently, a direct sum of two-sided ideals). Here is the group ring of the quaternion group over :

More generally, the decomposition works over a uniquely 2-divisible ring where all the irreducible representations can be realized:

Note that this does not work over a ring such as . Although all irreducible representations can be realized over this ring, the absence of unique 2-divisibility is a roadblock.

Explicit decomposition and idempotents for a splitting field

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For a non-splitting field

If is not a splitting field, then we get a decomposition of the form:

where is a subalgebra of that has dimension four over , and looks like a quaternion algebra.

Explicit decomposition and idempotents for a non-splitting field

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Relation with subgroups

Induced representations from subgroups

Since the quaternion 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 ) induces the double of the two-dimensional irreducible representation of the quaternion group.
  3. The trivial representation on the cyclic subgroup generated by induces a representation on the whole group that is the sum of a trivial representation and the representation with the -kernel. Analogous statements hold for .
  4. A representation on that sends to induces a representation of the whole group that is the sum of the representations with -kernel and -kernel. Analogous statements hold for .
  5. A representation on that sends to (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 of cyclic subgroups span the space of class functions. This is easy to check for the quaternion group from the points made above. By point (2) or point (5), the two-dimensional irreducible character is in the span. Points (3) and (4) show that all pairwise sums of the four one-dimensional representations are in the span. Taking suitable linear combinations of these yields that all the four one-dimensional representations are in the span.