Linear representation theory of general affine group of degree one over a finite field
From Groupprops
This article gives specific information, namely, linear representation theory, about a family of groups, namely: general affine group of degree one.
View linear representation theory of group families | View other specific information about general affine group of degree one
This article discusses the linear representation theory of the general affine group of degree one over a finite field of size and characteristic
, where
.
Summary
Item | Value |
---|---|
degrees of irreducible representations over a splitting field | 1 (![]() ![]() |
number of irreducible representations | ![]() See number of irreducible representations equals number of conjugacy classes, element structure of general affine group of degree one over a finite field |
maximum degree of irreducible representation over a splitting field | ![]() |
lcm of degrees of irreducible representations over a splitting field | ![]() |
sum of squares of degrees of irreducible representations | ![]() sum of squares of degrees of irreducible representations equals group order |
condition for a field to be a splitting field | The field should be a splitting field for the multiplicative group and its characteristic should not be ![]() ![]() ![]() ![]() ![]() |
smallest size splitting field | The finite field whose size is the smallest prime power other than ![]() ![]() |
Particular cases
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Group ![]() |
Order of the group | Second part of GAP ID | Degrees of irreducible representations | Number of irreducible representations | Linear representation theory page |
---|---|---|---|---|---|---|---|
2 | 2 | cyclic group:Z2 | 2 | 1 | 1,1 | 2 | linear representation theory of cyclic group:Z2 |
3 | 3 | symmetric group:S3 | 6 | 1 | 1,1,2 | 3 | linear representation theory of symmetric group:S3 |
4 | 2 | alternating group:A4 | 12 | 3 | 1,1,1,3 | 4 | linear representation theory of alternating group:A4 |
5 | 5 | general affine group:GA(1,5) | 20 | 3 | 1,1,1,1,4 | 5 | linear representation theory of general affine group:GA(1,5) |
7 | 7 | general affine group:GA(1,7) | 42 | 1 | 1,1,1,1,1,1,6 | 7 | linear representation theory of general affine group:GA(1,7) |
8 | 2 | general affine group:GA(1,8) | 56 | 11 | 1,1,1,1,1,1,1,7 | 8 | linear representation theory of general affine group:GA(1,8) |
9 | 3 | general affine group:GA(1,9) | 72 | 39 | 1,1,1,1,1,1,1,1,8 | 9 | linear representation theory of general affine group:GA(1,9) |
Irreducible representations
We denote by the pair the element
. The multiplicative part is
and the additive part is
.
Description of collection of representations | Parameter for describing the representation | How the representation is described | Degree of each representation | Number of representations | Sum of squares of degrees |
---|---|---|---|---|---|
One-dimensional, depends only on multiplicative part, kernel contains additive subgroup | a homomorphism ![]() |
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1 | ![]() |
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Nontrivial component of permutation representation on ![]() |
None | Consider the permutation representation of ![]() ![]() ![]() |
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1 | ![]() |
Total | -- | -- | -- | ![]() |
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