Linear representation theory of projective general linear group of degree two over a finite field
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This article gives specific information, namely, linear representation theory, about a family of groups, namely: projective general linear group of degree two.
View linear representation theory of group families | View other specific information about projective general linear group of degree two
This article describes the linear representation theory of the projective general linear group of degree two over a finite field. The order (size) of the field is , and the characteristic prime is
.
is a power of
. The group is denoted
or
.
See also the linear representation theory for: special linear group, projective special linear group, and general linear group.
Summary
Item | Value |
---|---|
degrees of irreducible representations over a splitting field | Case ![]() ![]() ![]() ![]() ![]() ![]() Case ![]() ![]() ![]() ![]() ![]() ![]() |
number of irreducible representations | Case ![]() ![]() ![]() ![]() See number of irreducible representations equals number of conjugacy classes, element structure of projective general linear group of degree two over a finite field#Conjugacy class structure |
quasirandom degree (minimum degree of nontrivial ireducible representation) | 1 |
maximum degree of irreducible representation | ![]() |
lcm of degrees of irreducible representations | Case ![]() ![]() ![]() ![]() |
sum of squares of degrees of irreducible representations | ![]() |
Particular cases
Irreducible representations
Case
,
odd
Description of collection of representations | Parameter for describing each representation | How the representation is described | Degree of each representation | Number of representations | Sum of squares of degrees |
---|---|---|---|---|---|
Trivial | -- | ![]() |
1 | 1 | 1 |
Sign representation | -- | Kernel is projective special linear group of degree two, image is ![]() |
1 | 1 | 1 |
Unclear | a nontrivial homomorphism ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
unclear | ![]() |
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Nontrivial component of permutation representation of ![]() ![]() |
-- | -- | ![]() |
1 | ![]() |
Tensor product of sign representation and nontrivial component of permutation representation on projective line | -- | -- | ![]() |
1 | ![]() |
Induced from one-dimensional representation of Borel subgroup | ![]() ![]() ![]() ![]() |
Induced from the following representation of the image of the Borel subgroup: ![]() |
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Total | NA | NA | NA | ![]() |
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