# 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).
View linear representation theory of particular groups | View other specific information about general linear group:GL(2,3)

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.

## Summary

Item Value
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) $\mathbb{Z}[\sqrt{-2}]$, same as $\mathbb{Z}[t]/(t^2 + 2)$
field generated by character values (characteristic zero) $\mathbb{Q}(\sqrt{-2})$, same as $\mathbb{Q}[t]/(t^2 + 2)$
other groups having the same character table binary octahedral group, see linear representation theory of binary octahedral group.

## Irreducible representations

### 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 $q$) Degree of each representation ($q = 3$) Number of representations (generic $q$) Number of representations ($q = 3$) Sum of squares of degrees (generic $q$) Sum of squares of degrees ($q = 3$)
One-dimensional, factor through the determinant map a homomorphism $\alpha: \mathbb{F}_3^\ast \to \mathbb{C}^\ast$ $x \mapsto \alpha(\det x)$ 1 1 $q - 1$ 2 $q - 1$ 2
Unclear a homomorphism $\varphi:\mathbb{F}_{9}^\ast \to \mathbb{C}^\ast$ unclear $q - 1$ 2 $q(q - 1)/2$ 3 $q(q - 1)^3/2$ 12
Tensor product of one-dimensional representation and the nontrivial component of permutation representation of $GL_2$ on the projective line over $\mathbb{F}_3$ a homomorphism $\alpha: \mathbb{F}_3^\ast \to \mathbb{C}^\ast$ $x \mapsto \alpha(\det x)\nu(x)$ where $\nu$ is the nontrivial component of permutation representation of $GL_2$ on the projective line over $\mathbb{F}_3$ $q$ 3 $q - 1$ 2 $q^2(q - 1)$ 18
Induced from one-dimensional representation of Borel subgroup Both distinct representations $\alpha, \beta$ homomorphisms $\mathbb{F}_3^\ast \to \mathbb{C}^\ast$ Induced from the following representation of the Borel subgroup: $\begin{pmatrix} a & b \\ 0 & d \\\end{pmatrix} \mapsto \alpha(a)\beta(d)$ $q + 1$ 4 $(q - 1)(q - 2)/2$ 1 $(q+1)^2(q-1)(q-2)/2$ 16
Total NA NA NA NA $q^2 - 1$ 8 $q^4 - q^3 - q^2 + q$ 48

## Character table

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 $\sqrt{-2}$ a fixed square root of -2.

Representation/conjugacy class representative and size $\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix}$ (size 1) $\begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix}$ (size 1) $\begin{pmatrix} 0 & 1 \\ -1 & 0 \\\end{pmatrix}$ (size 6) $\begin{pmatrix} 0 & 1 \\ 1 & -1 \\\end{pmatrix}$ (size 6) $\begin{pmatrix} 0 & 1 \\ 1 & 1 \\\end{pmatrix}$ (size 6) $\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix}$ (size 8) $\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix}$ (size 8) $\begin{pmatrix} 1 & 0 \\ 0 & -1 \\\end{pmatrix}$ (size 12)
trivial 1 1 1 1 1 1 1 1
nontrivial one-dimensional 1 1 1 -1 -1 1 1 -1
two-dimensional (unclear) 2 2 2 0 0 -1 -1 0
two-dimensional (unclear) 2 -2 0 $\sqrt{-2}$ $-\sqrt{-2}$ -1 1 0
two-dimensional (unclear) 2 -2 0 $-\sqrt{-2}$ $\sqrt{-2}$ -1 1 0
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 $S_4$ 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