Linear representation theory of finite cyclic groups
This article gives specific information, namely, linear representation theory, about a family of groups, namely: cyclic group.
View linear representation theory of group families | View other specific information about cyclic group
This article discusses the linear representation theory of a finite cyclic group of order , which for concreteness we take as the group of integers modulo n, i.e., the group
or
.
For the linear representation theory of the infinite cyclic group, see linear representation theory of group of integers.
Summary
We denote by the Euler totient function of
, defined as the number of numbers in
that are relatively prime to
, or equivalently as the order of the multiplicative group of the ring of integers modulo
.
Item | Value |
---|---|
degrees of irreducible representations over a splitting field | 1 (![]() maximum: 1, lcm: 1, number: ![]() ![]() |
Schur index values of irreducible representations | 1 (![]() |
smallest ring of realization of all irreducible representations | ![]() ![]() ![]() ![]() ![]() |
minimal splitting field (smallest field of realization of all irreducible representations) in characteristic zero | ![]() ![]() ![]() ![]() ![]() |
condition for field to be a splitting field | characteristic does not divide ![]() ![]() |
minimal splitting field in characteristic ![]() ![]() |
Field of size ![]() ![]() ![]() |
smallest size splitting field | smallest prime power ![]() ![]() ![]() |
degrees of irreducible representations over the reals | Case ![]() ![]() Case ![]() ![]() |
degrees of irreducible representations over the rationals | For every positive divisor ![]() ![]() ![]() ![]() maximum: ![]() ![]() ![]() ![]() |
Particular cases
![]() |
cyclic group | linear representation theory page | degrees of irreducible representations over splitting field | degrees of irreducible representations over reals | degrees of irreducible representations over rationals | smallest splitting field (characteristic zero) | smallest size of finite splitting field |
---|---|---|---|---|---|---|---|
1 | trivial group | -- | 1 | 1 | 1 | ![]() |
2, i.e., field:F2 |
2 | cyclic group:Z2 | linear representation theory of cyclic group:Z2 | 1,1 | 1,1 | 1,1 | ![]() |
3, i.e., field:F3 |
3 | cyclic group:Z3 | linear representation theory of cyclic group:Z3 | 1,1,1 | 1,2 | 1,2 | ![]() ![]() |
4, i.e., field:F4 |
4 | cyclic group:Z4 | linear representation theory of cyclic group:Z4 | 1,1,1,1 | 1,1,2 | 1,1,2 | ![]() ![]() ![]() |
5, i.e., field:F5 |
5 | cyclic group:Z5 | linear representation theory of cyclic group:Z5 | 1,1,1,1,1 | 1,2,2 | 1,4 | ![]() ![]() |
11, i.e., field:F11 |
6 | cyclic group:Z6 | linear representation theory of cyclic group:Z6 | 1,1,1,1,1,1 | 1,1,2,2 | 1,1,2,2 | ![]() ![]() |
7, i.e., field:F7 |
7 | cyclic group:Z7 | linear representation theory of cyclic group:Z7 | 1,1,1,1,1,1,1 | 1,2,2,2 | 1,6 | ![]() ![]() |
8, i.e., field:F8 |
8 | cyclic group:Z8 | linear representation theory of cyclic group:Z8 | 1,1,1,1,1,1,1,1 | 1,1,2,2,2 | 1,1,2,4 | ![]() ![]() |
9, i.e., field:F9 |
9 | cyclic group:Z9 | linear representation theory of cyclic group:Z9 | 1 (9 times) | 1,2,2,2,2 | 1,2,6 | ![]() ![]() |
19, i.e., field:F19 |
Group ring interpretation
For any commutative unital ring , the group ring
is isomorphic to the ring:
where the isomorphism sends an element to the equivalence class of
.
Case of ring that contains primitive roots of unity and is uniquely
-divisible
If contains a primitive
root of unity, say
, then we have:
If is uniquely
-divisible, then by the Chinese remainder theorem, we get the following direct sum/direct product decomposition as rings:
(We could also use instead of
in the line above).
The choice of decomposition using the Chinese remainder theorem also corresponds to the use of characters, as follows. We have an isomorphism:
where the are idempotents. Each idempotent is given as follows: take a character, and consider the element of the group ring obtained by multiplying each group element by its character value, and dividing the sum by the order of the group. The idempotents look like:
Note that in the case of fields, unique -divisibility is equivalent to saying that the characteristic of the field is either zero or is a prime not dividing
.
Also note that the decomposition does not work over , because of the absence of
-divisibility. This is true even though the representations can be realized over
.