Linear representation theory of 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: general linear group of degree two. This article restricts attention to the case where the underlying ring is a finite field.
View linear representation theory of group families | View other specific information about general linear group of degree two | View other specific information about group families for rings of the type finite field
This article describes the linear representation theory of the 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
.
See also the linear representation theories of: special linear group of degree two, projective general linear group of degree two, and projective special linear group of degree two.
Summary
Item | Value |
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degrees of irreducible representations over a splitting field | 1 (![]() ![]() ![]() ![]() ![]() ![]() ![]() |
number of irreducible representations | ![]() |
maximum degree of irreducible representation over a splitting field | ![]() |
lcm of degrees of irreducible representations over a splitting field | Case ![]() ![]() ![]() ![]() |
sum of squares of degrees of irreducible representations over a splitting field | ![]() |
Particular cases
Group | ![]() |
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Order of the group ![]() |
Number of irreducible representations ![]() |
Degrees of irreducible representations | Linear representation theory page |
---|---|---|---|---|---|---|
symmetric group:S3 | 2 | 2 | 6 | 3 | 1,1,2 | linear representation theory of symmetric group:S3 |
general linear group:GL(2,3) | 3 | 3 | 48 | 8 | 1,1,2,2,2,3,3,4 | linear representation theory of general linear group:GL(2,3) |
direct product of A5 and Z3 | 2 | 4 | 180 | 15 | 1,1,1,3,3,3,3,3,3,4,4,4,5,5,5 | |
general linear group:GL(2,5) | 5 | 5 | 480 | 24 | 1 (4 times), 4 (10 times), 5 (4 times), 6 (6 times) | linear representation theory of general linear group:GL(2,5) |
general linear group:GL(2,7) | 7 | 7 | 2016 | 48 | 1 (6 times), 6 (21 times), 7 (6 times), 8 (15 times) | linear representation theory of general linear group:GL(2,7) |
Irreducible representations
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 |
---|---|---|---|---|---|
One-dimensional, factor through the determinant map | a homomorphism ![]() |
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1 | ![]() |
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Unclear | a homomorphism ![]() ![]() ![]() |
unclear | ![]() |
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Tensor product of one-dimensional representation and the nontrivial component of permutation representation of ![]() ![]() |
a homomorphism ![]() |
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Induced from one-dimensional representation of Borel subgroup | ![]() ![]() ![]() ![]() |
Induced from the following representation of the Borel subgroup: ![]() |
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Total | NA | NA | NA | ![]() |
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