Modular representation theory of special linear group:SL(2,3) at 3

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This article gives specific information, namely, modular representation theory, about a particular group, namely: special linear group:SL(2,3).
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This article describes the modular representation theory of special linear group:SL(2,3), i.e. ,the linear representation theory in characteristic three -- over field:F3 and its extensions.

Summary

Item Value
degrees of irreducible representations (or degrees of irreducible Brauer characters) 1,2,3
number: 3
smallest field of realization of all irreducible representations field:F3

Family contexts

Family name Parameter value General discussion of modular representation theory of family
special linear group of degree two field:F3, i.e., the field with three elements modular representation theory of special linear group of degree two over a finite field in its defining characteristic

Character table

Below is the character table in characteristic 3. The entries are given modulo 3, and are written as 0,1,2.

Representation/conjugacy class representative and size \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix} -- 3-regular, size 1 \begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix} -- 3-regular, size 1 \begin{pmatrix} 0 & -1 \\ 1 & 0 \\\end{pmatrix} -- 3-regular, size 6 \begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix} -- not 3-regular, 4 elements \begin{pmatrix} 1 & -1 \\ 0 & 1 \\\end{pmatrix} -- not 3-regular, 4 elements \begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix} -- not 3-regular, 4 elements \begin{pmatrix} -1 & -1 \\ 0 & -1 \\\end{pmatrix} -- not 3-regular, 4 elements
trivial 1 1 1 1 1 1 1
two-dimensional, given by identity mapping 2 1 0 2 2 1 1
three-dimensional, given by symmetric square of two-dimensional 0 0 2 0 0 0 0

Brauer characters

Brauer character table

Irreducible representation in characteristic three whose Brauer character we are computing Irreducible representation in characteristic zero whose character equals the Brauer character Value of Brauer character on conjugacy class of \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix} -- size 1 Value of Brauer character on conjugacy class of \begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix} -- size 1 Value of Brauer character on conjugacy class of \begin{pmatrix} 0 & -1 \\ 1 & 0 \\\end{pmatrix} -- size 6
trivial trivial 1 1 1
two-dimensional, given by identity mapping quaternionic representation of special linear group:SL(2,3) 2 -2 0
three-dimensional, given by symmetric square of two-dimensional kernel is center, reduces to standard representation of alternating group:A4 on quotient 3 3 -1