View source for Linear representation theory of quaternion group

{{group-specific information|
information type = linear representation theory|
group = quaternion group|
connective = of}}

==Summary==

<section begin="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:D8|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.

{| class="sortable" border="1"
! Item !! Value
|-
| [[Degrees of irreducible representations]] over a [[splitting field]] (such as <math>\mathbb{C}</math> or <math>\overline{\mathbb{Q}}</math>) || 1,1,1,1,2<br>[[maximum degree of irreducible representation|maximum]]: 2, [[lcm of degrees of irreducible representations|lcm]]: 2, [[number of irreducible representations equals number of conjugacy classes|number]]: 5, [[sum of squares of degrees of irreducible representations equals order of group|sum of squares]]: 8
|-
| [[Schur index]] values of irreducible representations || 1,1,1,1,2 (characteristic zero)<br>[[maximum Schur index of irreducible representation|maximum]]: 2, [[lcm of Schur indices of irreducible representations|lcm]]: 2<br>1,1,1,1,1 (characteristic other than 0,2)
|-
| Smallest ring of realization for all irreducible representations (characteristic zero) || There are multiple candidates. <math>\mathbb{Z}[i]</math> where <math>i</math> is a square root of <math>-1</math>, equivalently <math>\mathbb{Z}[t]/(t^2 + 1)</math>, the [[ring of Gaussian integers]] is one candidate. Another is <math>\mathbb{Z}[\sqrt{-2}]</math> or <math>\mathbb{Z}[t]/(t^2 + 2)</math>.<br>More generally, any ring of the form <math>\mathbb{Z}[\alpha,\beta]</math> where <math>\alpha^2 + \beta^2 = -1</math> is a ring of realization for all irreducible representations. In particular, <math>\mathbb{Z}[\sqrt{-m^2 - 1}]</math> works for any rational <math>m</math>.
|-
| [[Minimal splitting field]] (i.e., field of realization) for all irreducible representations (characteristic zero) || There are multiple candidates. <math>\mathbb{Q}(i)</math> or <math>\mathbb{Q}[t]/(t^2 + 1)</math> works, so does <math>\mathbb{Q}(\sqrt{2}i)</math> or <math>\mathbb{Q}[t]/(t^2 + 2)</math>. More generally, <math>\mathbb{Q}(\alpha,\beta)</math> where <math>\alpha^2 + \beta^2 = -1</math> is a splitting field. In particular, <math>\mathbb{Q}(\sqrt{-1-m^2})</math> works for any rational <math>m</math>.<br>See [[minimal splitting field need not be unique]], [[minimal splitting field need not be cyclotomic]]
|-
| [[Ring generated by character values]] (characteristic zero) || <math>\mathbb{Z}</math>
|-
| [[Field generated by character values]] (characteristic zero) || <math>\mathbb{Q}</math> (hence it is a [[rational group]])<br>See also: [[Field generated by character values need not be a splitting field]]<nowiki>|</nowiki>[[rational not implies rational-representation]]
|-
| Condition for being a [[splitting field]] for this group || Sufficient condition: the characteristic is not two and there exist <math>\alpha, \beta</math> in the field such that <math>\alpha^2 + \beta^2 + 1 = 0</math>. 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, <math>-1</math> is a sum of two squares in any finite field.
|- 
| [[Minimal splitting field]] (characteristic <math>p \ne 0,2</math>) || The [[prime field]] <math>\mathbb{F}_p</math>
|-
| 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)<br>[[number of orbits of irreducible representations equals number of orbits under automorphism group|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)<br>[[number of projective equivalence classes of irreducible linear representations|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<br>[[Number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy|number]]: 5
|-
| [[Character table-equivalent groups|Groups with same character table]] || [[Dihedral group:D8]]
|}
<section end="summary"/>

==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.

{| class="sortable" border="1"
! Name of representation type !! Number of representations of this type !! [[Degree of a linear representation|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]]: <math>\langle i \rangle</math>, <math>\langle j \rangle</math>, <math>\langle k \rangle</math> || [[cyclic group:Z2]] ||  works, same as trivial
|-
| [[faithful irreducible representation of quaternion group|two-dimensional irreducible]] || 1 || 2 || 2 || <math>-1</math> 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 field]]s 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.

{| class="sortable" border="1"
! Name of representation type !! Number of representations of this type !! [[Degree of a linear representation|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.

{| class="sortable" border="1"
! Element !! Matrix !! Characteristic polynomial !! Minimal polynomial !! Trace, character value
|-
| <math>1</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>-1</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>i</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>-i</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>j</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>-j</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>k</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>-k</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|}


===Sign representations with <math>i,j,k</math>-kernels===

The quaternion group has three [[maximal normal subgroup]]s: the cyclic subgroups generated by <math>i,j,k</math> 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 <math>\langle i \rangle</math>-kernel:

{| class="sortable" border="1"
! Element !! Matrix !! Characteristic polynomial !! Minimal polynomial !! Trace, character value, determinant (all same because it is degree 1)
|-
| <math>1</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>-1</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>i</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>-i</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>j</math> || <math>( -1 )</math> || <math>t + 1</math> || <math>t + 1</math> || -1
|-
| <math>-j</math> || <math>( -1 )</math> || <math>t + 1</math> || <math>t + 1</math> || -1
|-
| <math>k</math> || <math>( -1 )</math> || <math>t + 1</math> || <math>t + 1</math> || -1
|-
| <math>-k</math> || <math>( -1 )</math> || <math>t + 1</math> || <math>t + 1</math> || -1
|}

Representation with <math>\langle j \rangle</math>-kernel:

{| class="sortable" border="1"
! Element !! Matrix !! Characteristic polynomial !! Minimal polynomial !! Trace, character value, determinant (all same because it is degree 1)
|-
| <math>1</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>-1</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>i</math> || <math>( -1 )</math> || <math>t + 1</math> || <math>t + 1</math> || -1
|-
| <math>-i</math> || <math>( -1 )</math> || <math>t + 1</math> || <math>t + 1</math> || -1
|-
| <math>j</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>-j</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>k</math> || <math>( -1 )</math> || <math>t + 1</math> || <math>t + 1</math> || -1
|-
| <math>-k</math> || <math>( -1 )</math> || <math>t + 1</math> || <math>t + 1</math> || -1
|}

Representation with <math>\langle k \rangle</math>-kernel:

{| class="sortable" border="1"
! Element !! Matrix !! Characteristic polynomial !! Minimal polynomial !! Trace, character value, determinant (all same because it is degree 1)
|-
| <math>1</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>-1</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>i</math> || <math>( -1 )</math> || <math>t + 1</math> || <math>t + 1</math> || -1
|-
| <math>-i</math> || <math>( -1 )</math> || <math>t + 1</math> || <math>t + 1</math> || -1
|-
| <math>j</math> || <math>( -1 )</math> || <math>t + 1</math> || <math>t + 1</math> || -1
|-
| <math>-j</math> || <math>( -1 )</math> || <math>t + 1</math> || <math>t + 1</math> || -1
|-
| <math>k</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|-
| <math>-k</math> || <math>( 1 )</math> || <math>t - 1</math> || <math>t - 1</math> || 1
|}

===Two-dimensional irreducible representation over a splitting field===

{{further|[[Faithful irreducible representation of quaternion group]]}}

{{#lst:faithful irreducible representation of quaternion group|representation table}}

===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 <math>\{ 1,i,j,k \}</math>. Note that multiplication by anything other than <math>\pm 1</math> 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 <math>\pm 1</math>.

{| class="sortable" border="1"
! Element !! Matrix !! Characteristic polynomial !! Minimal polynomial !! Trace, character value
|-
| <math>1</math> || <math>\begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\\end{pmatrix}</math> || <math>(t - 1)^4 = t^4 - 4t^3 + 6t^2 - 4t + 1</math> || <math>t - 1</math> || 4
|-
| <math>-1</math> || <math>\begin{pmatrix}-1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\\end{pmatrix}</math> || <math>(t + 1)^4 = t^4 + 4t^3 + 6t^2 + 4t + 1</math> || <math>t + 1</math> || -4
|-
| <math>i</math> || <math>\begin{pmatrix}0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\\end{pmatrix}</math> || <math>(t^2 + 1)^2 = t^4 + 2t^2 + 1</math> || <math>t^2 + 1</math> || 0
|-
| <math>-i</math> || <math>\begin{pmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \\\end{pmatrix}</math> || <math>(t^2 + 1)^2 = t^4 + 2t^2 + 1</math> || <math>t^2 + 1</math> || 0
|-
| <math>j</math> || <math>\begin{pmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\\end{pmatrix}</math> || <math>(t^2 + 1)^2 = t^4 + 2t^2 + 1</math> || <math>t^2 + 1</math> || 0
|-
| <math>-j</math> || <math>\begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\\end{pmatrix}</math> || <math>(t^2 + 1)^2 = t^4 + 2t^2 + 1</math> || <math>t^2 + 1</math> || 0
|-
| <math>k</math> || <math>\begin{pmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\\end{pmatrix}</math> || <math>(t^2 + 1)^2 = t^4 + 2t^2 + 1</math> || <math>t^2 + 1</math> || 0
|-
| <math>-k</math> || <math>\begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \\\end{pmatrix}</math> || <math>(t^2 + 1)^2 = t^4 + 2t^2 + 1</math> || <math>t^2 + 1</math> || 0
|}

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

==Character table==

{{character table facts to check against}}

<section begin="character table"/>

This character table works over characteristic zero and over any other characteristic not equal to two once we reduce the entries mod the characteristic:

{| class="sortable" border="1"
!Representation/Conjugacy class !! <matH>\{ 1 \}</math> (identity; size 1) || <math>\{ -1 \}</math> (size 1) || <math>\{ i, -i \}</math> (size 2) || <math>\{ j, -j \}</math> (size 2)|| <math>\{ k, -k \}</math> (size 2)
|-
| [[Trivial representation]] || 1 || 1 || 1 || 1 || 1
|-
| <math>i</math>-kernel || 1 || 1 || 1 || -1 || -1
|-
| <math>j</math>-kernel || 1 || 1 || -1 || 1 || -1
|-
| <math>k</math>-kernel || 1 || 1 || -1 || -1 || 1
|-
| [[two-dimensional irreducible representation of quaternion group|2-dimensional]] || 2 || -2 || 0 || 0 || 0
|}

<section end="character table"/>
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]]:

{| class="sortable" border="1"
!Representation/Conjugacy class !! <matH>\{ 1 \}</math> (identity; size 1) || <math>\{ -1 \}</math> (size 1) || <math>\{ i, -i \}</math> (size 2) || <math>\{ j, -j \}</math> (size 2)|| <math>\{ k, -k \}</math> (size 2)
|-
| [[trivial representation]] || 1 || 1 || 2 || 2 || 2
|-
| <math>i</math>-kernel || 1 || 1 || 2 || -2 || -2
|-
| <math>j</math>-kernel || 1 || 1 || -2 || 2 || -2
|-
| <math>k</math>-kernel || 1 || 1 || -2 || -2 || 2
|-
| [[two-dimensional irreducible representation of quaternion group|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 <math>2</math>.

{| class="sortable" border="1"
! Type of field !! Condition !! Condition on <math>q</math> for finite field of size <math>q</math> !! 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===

{| class="sortable" border="1"
! Representation !! Smallest ring of realization !! Smallest set of elements occurring as entries of matrices
|-
| trivial representation || <math>\mathbb{Z}</math> -- ring of integers || <math>\{ 1 \}</math>
|-
| <math>i</math>-kernel || <math>\mathbb{Z}</math> -- ring of integers || <math>\{ 1, -1 \}</math>
|-
| <math>j</math>-kernel || <math>\mathbb{Z}</math> -- ring of integers || <math>\{ 1, -1 \}</math>
|-
| <math>k</math>-kernel || <math>\mathbb{Z}</math> -- ring of integers || <math>\{ 1, -1 \}</math>
|-
| two-dimensional irreducible (over splitting field) || <math>\mathbb{Z}[i]</math>; <math>\mathbb{Z}[\sqrt{2}i]</math>; <math>\mathbb{Z}[\alpha,\beta]</math> where <math>\alpha^2 + \beta^2 = -1</math> || <math>\{ 0,1,-1,i,-i \}</math> if over <math>\mathbb{Z}[i]</math>; <math>\{ 0,1,-1,\sqrt{2}i,-\sqrt{2}i \}</math> if over <math>\mathbb{Z}[\sqrt{2}i]</math>; <math>\{0,1,-1,\alpha,-\alpha,\beta,-\beta\}</math> if over <math>\mathbb{Z}[\alpha,\beta]</math>
|-
| four-dimensional irreducible (over non-splitting field) || <math>\mathbb{Z}</math> -- ring of integers || <math>\{ 0,1,-1 \}</math>
|}

==Orthogonality relations and numerical checks==

{| class="sortable" border="1"
! 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 <math>n^{th}</math> roots where <math>n</math> is the exponent of the group, it is a splitting field. || In this case, <math>n = 4</math>, and having a primitive fourth root is equivalent to <math>-1</math> 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 <math>\{ 1, -1 \}</math> (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]] || <math>1^2 + 1^2 + 1^2 + 1^2 + 2^2 = 8</math>.
|-
| [[degree of irreducible representation divides index of abelian normal subgroup]] || All degrees <math>1,1,1,1,2</math> divide 2, which is the index of the abelian normal subgroup <math>\langle i \rangle</math> (and also of <math>\langle j \rangle</math> and <math>\langle k \rangle</math>.
|-
| [[order of inner automorphism group bounds square of degree of irreducible representation]] || The [[center]] is <math>\{ 1, -1 \}</math> (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==

{{irrep degrees automorphism group action facts to check against}}

===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:

[[File:Q8latticeofsubgroups.png|500px]]

{| class="sortable" border="1"
! 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 (<math>\langle i \rangle</math>-kernel, <math>\langle j \rangle</math>-kernel, and <math>\langle k \rangle</math>-kernel sign depending on which subgroup is chosen) || 1 || sign (<math>\langle i \rangle</math>-kernel, <math>\langle j \rangle</math>-kernel, and <math>\langle k \rangle</math>-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 <math>\langle i \rangle</math>, <matH>\langle j \rangle</math>, and <math>\langle k \rangle</math> 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|[[Group ring over splitting field is direct sum of matrix rings for each irreducible representation]]}}

Over a splitting field <math>K</math> (which could be a finite field or a field such as <math>\mathbb{Q}[i]</math> or <math>\mathbb{C}</math>), we have the decomposition (as a direct product of rings, or equivalently, a direct sum of two-sided ideals). Here <math>K[G]</math> is the [[group ring]] of the quaternion group over <math>K</math>:

<math>K[G] \cong M_1(K) \oplus M_1(K) \oplus M_1(K) \oplus M_1(K) \oplus M_2(K) \cong K \oplus K \oplus K \oplus K \oplus M_2(K)</math>

More generally, the decomposition works over a ''uniquely 2-divisible'' ring <math>R</math> where all the irreducible representations can be realized:

<math>R[G] \cong M_1(R) \oplus M_1(R) \oplus M_1(R) \oplus M_1(R) \oplus M_2(R) \cong R \oplus R \oplus R \oplus R \oplus M_2(R)</math>

Note that this does ''not'' work over a ring such as <math>\mathbb{Z}[i]</math>. 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===

{{fillin}}

===For a non-splitting field===

If <math>K</math> is not a splitting field, then we get a decomposition of the form:

<math>K[G] \cong M_1(K) \oplus M_1(K) \oplus M_1(K) \oplus M_1(K) \oplus L</math>

where <math>L</math> is a subalgebra of <math>M_4(K)</math> that has dimension four over <math>K</math>, and looks like a quaternion algebra.

===Explicit decomposition and idempotents for a non-splitting field===

{{fillin}}

==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).

# The trivial representation on the center induces a representation obtained as a sum of the four one-dimensional representations.
# The sign representation on the center (which comprises <math>\pm 1</math>) induces the double of the two-dimensional irreducible representation of the quaternion group.
# The trivial representation on the cyclic subgroup generated by <math>i</math> induces a representation on the whole group that is the sum of a trivial representation and the representation with the <math>i</math>-kernel. Analogous statements hold for <math>j,k</math>.
# A representation on <math>\langle i \rangle</math> that sends <math>i</math> to <math>-1</math> induces a representation of the whole group that is the sum of the representations with <math>j</math>-kernel and <math>k</math>-kernel. Analogous statements hold for <math>j,k</math>.
# A representation on <math>\langle i \rangle</math> that sends <math>i</math> to <math>i</math> (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.