Linear representation theory of general linear group:GL(2,3)
This article gives specific information, namely, linear representation theory, about a particular group, namely: general linear group:GL(2,3).
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This article describes the linear representation theory (in characteristic zero and other characteristics excluding 2,3) of general linear group:GL(2,3), which is the general linear group of degree two over field:F3.
|degrees of irreducible representations over a splitting field|| 1,1,2,2,2,3,3,4|
maximum: 4, lcm: 12, number: 8, sum of squares: 48
|ring generated by character values (characteristic zero)||, same as|
|field generated by character values (characteristic zero)||, same as|
|other groups having the same character table||binary octahedral group, see linear representation theory of binary octahedral group.|
Interpretation as general linear group of degree two
The group is a general linear group of degree two over field:F3. Compare with linear representation theory of general linear group of degree two over a finite field.
|Description of collection of representations||Parameter for describing each representation||How the representation is described||Degree of each representation (generic )||Degree of each representation ()||Number of representations (generic )||Number of representations ()||Sum of squares of degrees (generic )||Sum of squares of degrees ()|
|One-dimensional, factor through the determinant map||a homomorphism||1||1||2||2|
|Tensor product of one-dimensional representation and the nontrivial component of permutation representation of on the projective line over||a homomorphism||where is the nontrivial component of permutation representation of on the projective line over||3||2||18|
|Induced from one-dimensional representation of Borel subgroup||Both distinct representations homomorphisms||Induced from the following representation of the Borel subgroup:||4||1||16|
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
In the table below, we denote by a fixed square root of -2.
|Representation/conjugacy class representative and size||(size 1)||(size 1)||(size 6)||(size 6)||(size 6)||(size 8)||(size 8)||(size 12)|
|three-dimensional, factors through standard representation of symmetric group:S4||3||3||-1||-1||-1||0||0||1|
|three-dimensional, factors through tensor product of standard and sign representations of||3||3||-1||1||1||0||0||-1|
|four-dimensional, induced from one-dimensional representation of Borel subgroup||4||-4||0||0||0||1||-1||0|