nLab
GL(2,3)

Contents

Contents

Idea

The finite general linear group GL(2,𝟛)GL(2,\mathbb{3}) for coefficients the prime field 𝔽 3\mathbb{F}_3.

Properties

Character table

The character table for GL(2,𝔽 3)GL(2,\mathbb{F}_3) over the complex numbers looks like that of the binary octahedral group. The only difference is the Schur indices. Hence the character tables over the real numbers do differ.

linear representation theory of the finite general linear group GL(2,3)

\,

group order: |GL(2,3)|=48{\vert GL(2,3)\vert} = 48

conjugacy classes:12A2B3468A8B
their cardinality:111286866

character table over the complex numbers \mathbb{C}

irrep12A2B3468A8B
ρ 1\rho_111111111
ρ 2\rho_211-1111-1-1
ρ 3\rho_3220-12-100
ρ 4\rho_42-20-1012-\sqrt{-2}2\sqrt{-2}
ρ 5\rho_52-20-1012\sqrt{-2}2-\sqrt{-2}
ρ 6\rho_633-10-1011
ρ 7\rho_73310-10-1-1
ρ 8\rho_84-4010-100

character table over the rational numbers \mathbb{Q} and real numbers \mathbb{R}

irrep12A2B3468A8B
ρ 1\rho_111111111
ρ 2\rho_211-1111-1-1
ρ 3\rho_3220-12-100
ρ 4ρ 5\rho_4 \oplus \rho_54-40-20200
ρ 6\rho_633-10-1011
ρ 7\rho_73310-10-1-1
ρ 8\rho_84-4010-100

References

References

Created on October 6, 2018 at 07:35:20. See the history of this page for a list of all contributions to it.