Linear representation theory of cyclic group:Z3
This article gives specific information, namely, linear representation theory, about a particular group, namely: cyclic group:Z3.
View linear representation theory of particular groups | View other specific information about cyclic group:Z3
This article discusses the linear representation theory of cyclic group:Z3, a group of order three.
|Degrees of irreducible representations over a splitting field|| 1, 1, 1|
maximum: 1, lcm: 1, number: 3, sum of squares: 3
|Smallest ring of realization of all representations (characteristic zero)||, also expressible as . Degree two integral extension of .|
|Smallest field of realization of all representations (characteristic zero)||, also expressible as . Degree two cyclotomic extension of .|
|Criterion for a field to be a splitting field||Any field of characteristic not 3 that contains a primitive cube root of unity, i.e., the polynomial splits.|
|Degrees of irreducible representations over a non-splitting field. Includes the case of field of real numbers , also field of rational numbers|| 1, 2|
maximum: 2, lcm: 2, number: 2
|Smallest size splitting field||field:F4, i.e., field with four elements.|
|Family||Parameter value||General discussion of linear representation theory of family|
|finite cyclic group||3||linear representation theory of finite cyclic groups|
|alternating group||3||linear representation theory of alternating groups|
Below is summary information on irreducible representations. Note that a particular representation may make sense, and be irreducible, only for certain kinds of fields -- see the "Values not allowed for field characteristic" and "Criterion for field" columns to see the condition the field must satisfy for the representation to be irreducible there.
|Name of representation type||Number of representations of this type||Values not allowed for field characteristic||Criterion for field||What happens over a splitting field?||Kernel||Degree||Schur index||What happens by reducing the -representation over bad characteristics?|
|trivial||1||--||any||remains the same||whole group||1||1||--|
|one-dimensional nontrivial||2||3||contains a primitive cube root of unity||remains the same||trivial subgroup, i.e., it is faithful||1||1||irrelevant question|
|two-dimensional irreducible||1||--||does not contain a primitive cube root of unity||splits into the two one-dimensional nontrivial representations||trivial subgroup, i.e., it is faithful||2||1||Taking the -representation and mapping to field:F3 gives an indecomposable two-dimensional representation that is not irreducible.|
This is a one-dimensional representation sending all the elements to , and makes sense over any field.
One-dimensional nontrivial representations
The cyclic group of order three has two non-identity elements. Also, in a field with a primitive cube root of unity, there are two such primitive cube roots of unity.
The two one-dimensional nontrivial representations both send the identity element to and the two non-identity elements to the two primitive cube roots of unity. The representations differ in terms of which element of the group is matched with which primitive cube root of unity.
Two-dimensional representation: irreducible in the non-splitting case
If we denote the group as where is the identity element, we can construct the following representation with integer matrices:
We can interpret this representation over any field , and three cases arise as to the nature of :
|Case for the field||What happens in this case|
|does not have characteristic does not contain a primitive cube root of unity||The representation is irreducible. However, it is not absolutely irreducible, it decomposes as a direct sum in a quadratic extension containing a primitive cube root of unity (See next point).|
|does not have characteristic and contains a primitive cube root of unity||The representation decomposes as a direct sum of the two one-dimensional nontrivial representations with kernel V4 in A4|
|Characteristic of is||The representation is indecomposable but not irreducible, because there is an invariant one-dimensional subspace.|
Our case of current interest is the first one, i.e., where does not have characteristic and does not contain a primitive cube root of unity.
In the case is the field is the field of real numbers, or more generally, any subfield of the reals containing , we can provide an alternative description of this representation as rotations by multiples of . Note that this alternative description, though it gives an equivalent representation, does not work over fields that lack a square root of .
Character table over a splitting field
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
Let be a primitive cube root of unity. The character table over a splitting field is as follows:
|Representation/Conjugacy class||(identity element)||(generator)||(generator)|
|one nontrivial representation||1|
|the other (conjugate) nontrivial representation||1|
Note that this character table is interpreted differently depending on what the splitting field is and which of the primitive cube roots we choose to be . Switching the roles of and in the above table simply permutes the two nontrivial one-dimensional representations and has no effect on the overall character table.
In characteristic zero, can be taken as or , which is . is the other primitive cube root of unity, and is given as or or .
Character table over a non-splitting field
For a field that is not a splitting field for the group, there are only two equivalence classes of irreducible representations. But also, the number of Galois conjugacy classes is two. The character table looks as follows:
|Representation/Galois conjugacy class||Identity element||Non-identity elements|
|nontrivial two-dimensional representation||2||-1|
Over a finite field, the character values are interpreted as integers modulo the field characteristic; over an infinite field, they are interpreted as rational numbers and hence field elements.
If doing character theory over the real numbers, we know that the number of irreducible representations over reals equals number of real conjugacy classes. The above is the character table both over the rationals and over the reals.