Quaternionic representation of special linear group:SL(2,3)

From Groupprops
Jump to: navigation, search
This article describes a particular irreducible linear representation for the following group: special linear group:SL(2,3). The representation is unique up to equivalence of linear representations and is irreducible, at least over its original field of definition in characteristic zero. The representation may also be definable over other characteristics by reducing the matrices modulo that characteristic, though it may behave somewhat differently in these characteristics.
For more on the linear representation theory of the group, see linear representation theory of special linear group:SL(2,3).

Summary

Item Value
Degree of representation 2
Schur index 2 if the characteristic is 0, 1 in other characteristics
Kernel of representation trivial subgroup, i.e., it is a faithful linear representation (except in bad characteristics)
Set of character values \{ 0, 1, -1, 2, -2 \}
Characteristic zero: Ring generated: \mathbb{Z}, ring of integers; Ideal within ring generated: whole ring; Field generated: \mathbb{Q}, field of rational numbers
Rings of realization The representation can definitely be realized over any ring containing a primitive cube root of unity, such as \mathbb{Z}[t]/(t^2 + t + 1). There are probably other rings too?
Fields of realization (Need to determine a sufficient condition). In particular, the representation can be realized over any finite field (though it behaves differently in characteristic two).
Minimal field of realization In characteristic zero: prime field \mathbb{F}_p
In characteristic p: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
Type of representation Quaternionic, i.e., the character is real-valued but the representation cannot be realized over the reals. See the #Frobenius-Schur indicator section.
Size of equivalence class under automorphisms 1
Size of equivalence class under Galois automorphisms 1
Size of equivalence class under action of multiplicative group of one-dimensional representations by multiplication 1 if the field does not contain a primitive cube root of unity
3 if the field contains a primitive cube root of unity
Bad characteristic(s) 2.
Note that characteristic 3 is, surprisingly, not a bad characteristic -- we can take the natural embedding of SL(2,3) in GL(2,3) and get the irreducible representation in characteristic 3.

Representation table

PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Frobenius-Schur indicator

For background reference, see indicator theorem, indicator character, and Frobenius-Schur indicator

The Frobenius-Schur indicator of a representation is the inner product of the character of the representation and the indicator character (which is the character that assigns to every element is number of square roots). The Frobenius-Schur indicator can be computed as \sum \chi(g^2) where \chi is the character of the representation.

Note that for two elements in the same conjugacy class, their squares are also in the same conjugacy class as each other (though not necessarily the same conjugacy class as the original element). It thus suffices to compute \chi(g^2) for one element in each conjugacy class and multiply by the size of the conjugacy class.

Conjugacy class representative Size of conjugacy class Conjugacy class representative for square Character value at square (note: this is the trace of the image of the matrix under the representation. It so happens that because the SL(2,3) in GL(2,3) embedding is the realization of the representation in characteristic 3, it is also equal to the trace of the matrices mod 3) Total contribution of conjugacy class (= size of conjugacy class times character value of square)
\begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix} 1 \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix} 2 2
\begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix} 1 \begin{pmatrix} 1 & 0 \\ 0 & 1 \\\end{pmatrix} 2 2
\begin{pmatrix} 0 & -1 \\ 1 & 0 \\\end{pmatrix} 6 \begin{pmatrix} -1 & 0 \\ 0 & -1 \\\end{pmatrix} -2 -12
\begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix} 4 \begin{pmatrix} 1 & -1 \\ 0 & 1 \\\end{pmatrix} -1 -4
\begin{pmatrix} 1 & -1 \\ 0 & 1 \\\end{pmatrix} 4 \begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix} -1 -4
\begin{pmatrix} -1 & 1 \\ 0 & -1 \\\end{pmatrix} 4 \begin{pmatrix} 1 & 1 \\ 0 & 1 \\\end{pmatrix} -1 -4
\begin{pmatrix}- 1 & -1 \\ 0 & -1 \\\end{pmatrix} 4 \begin{pmatrix} 1 & -1 \\ 0 & 1 \\\end{pmatrix} -1 -4
Total 24 -- -- -24

The Frobenius-Schur indicator is thus -24/24 = -1, indicating that the representation has a real character but cannot be realized over the field of real numbers.