Projective Representations: a simple example

[rocdoc]

Could anyone help with providing a simple example of a projective representation of a small finite group ( order of group not greater than six )?

My understanding is that, if the group has N elements, then I should see N matrices in the projective representation.

I would prefer the example to not involve a direct prduct group.

I have used nice simple groups, like C_2v, in chemistry, this group has four elements.

 

[fresh_42]

Could anyone help with providing a simple example of a projective representation of a small finite group ( order of group not greater than six )?

My understanding is that, if the group has N elements, then I should see N matrices in the projective representation.

I would prefer the example to not involve a direct prduct group.

I have used nice simple groups, like C_2v, in chemistry, this group has four elements.

What do you mean by a projective representation? A (surjective) projection $G \longrightarrow GL(V)$ or a projective resolution of $G$ or $GL(V)$?

 

[martinbn]

What do you mean by a projective representation? A (surjective) projection $G \longrightarrow GL(V)$ or a projective resolution of $G$ or $GL(V)$?

https://en.wikipedia.org/wiki/Projective_representation

 

[rocdoc]

Hopefully the following helps with your question.

If the group is denoted by G and

$$g_1~ and ~g_2 $$are in G then we have, a projective representation of G if

$$M(g_1)M(g_2)=M(g_1g_2)e^{i\theta(g_1,g_2)}$$

With M meaning matrix.

 

[rocdoc]

See EQ(2) , pg3 of

https://arxiv.org/abs/1605.05805

 

[DrDu]

In chemistry, the projective representations raise their head in the form of so called "double groups". There, the projective factors are avoided by introducing another group element "R". Also the permutation groups S_n have projective representations for S_n when n>=4.

 

[fresh_42]

The smallest example which I have found is
$$
\mathbb{Z}_2\, , \,\mathbb{Z}_3\; , \; \mathbb{Z}_2 \rtimes \mathbb{Z}_3 \longrightarrow PSL(2,\mathbb{F}_2)=SL(2,\mathbb{F}_2) = \mathcal{Sym}(3) = D_3 = \mathbb{Z}_2 \rtimes \mathbb{Z}_3
$$
With less finiteness conditions on the group it might be worth mentioning
$$
\mathbb{S}^1 \cong SO(2,\mathbb{R}) \cong U(1,\mathbb{C}) \cong \mathbb{P}(1,\mathbb{R}) \text{ (unit circle) } \\ \text{ and } \\ \mathbb{S}^2 \cong SO(3,\mathbb{R})/SO(2,\mathbb{R}) \cong \mathbb{P}(1,\mathbb{C}) \text{ (Riemann sphere) }
$$
and finite subgroups (reflections, rotations) of the unit circle or the 2-sphere will provide examples.

 

[fresh_42]

An interesting example (https://dl.acm.org/citation.cfm?id=676246) of a projective representation can be found for the Lie algebra $\mathfrak{g}= \mathfrak{sl}(2,\mathbb{F})$ with $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$ but only because I don't remember the characteristic $2$ case and the centers need more care if other fields are involved. However, I assume that finite fields would work as well.

If we define the group $\Gamma^°(\mathfrak{g})$ of all $\phi^* \otimes \psi^* \otimes \chi \in GL(\mathfrak{g}^*\otimes \mathfrak{g}^* \otimes \mathfrak{g})$ such that $[X,Y]= \chi([\phi(X),\psi(Y)])$ for all $X,Y \in \mathfrak{g}$ then
$$
\Gamma^°(\mathfrak{g}) \cong PSL(2,\mathbb{F})
$$
which means the isometries of the Lie multiplication in $\mathfrak{sl}(2,\mathbb{F})$ have such a representation.

 

[rocdoc]

It looks like the Klein four-group, V₄ has a projective representation.

 

[rocdoc]

See at
https://groupprops.subwiki.org/wiki/Projective_representation_theory_of_Klein_four-group

 

[fresh_42]

It looks like the Klein four-group, V₄ has a projective representation.

Yes, but you ruled out this example:

I would prefer the example to not involve a direct prduct group.

 

[rocdoc]

The group multiplication table of V₄ can be given as

$$\begin{array}{|c|c|c|c|c|}

\hline & e & a & b & c \\

\hline e & e & a & b & c \\

\hline a & a & e & c & b \\

\hline b & b & c & e & a \\

\hline c & c & b & a & e \\

\hline

\end{array} $$
If we have four matrices M(e),M(a),M(b),M(c) representing our group elements e,a,b,c, respectively.Then they form a matrix representation of our group if the "matrix multiplication table" has the same structure as the group multiplication table. I.E if we have

$$\begin{array}{|c|c|c|c|c|}

\hline & M\left( e \right) & M\left( a \right) & M\left( b \right) & M\left( c \right) \\

\hline M\left( e \right) & M\left( e \right) & M\left( a \right) & M\left( b \right) & M\left( c \right) \\

\hline M\left( a \right) & M\left( a \right) & M\left( e \right) & M\left( c \right) & M\left( b \right) \\

\hline M\left( b \right) & M\left( b \right) & M\left( c \right) & M\left( e \right) & M\left( a \right) \\

\hline M\left( c \right) & M\left( c \right) & M\left( b \right) & M\left( a \right) & M\left( e \right) \\ \hline \end{array}

$$
Where the matrix in a cell of the table body, is formed by multiplying on the left, the matrix shown at the top of a particular column, by the matrix shown in the leftmost column for a particular row, in the natural obvious way.

If we use the following matrices

$$M(e)=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix},M(a)=\begin{bmatrix}0 & -1 \\1 & 0 \end{bmatrix},M(b)=\begin{bmatrix}1 & 0 \\0 & -1 \end{bmatrix},M(c)=\begin{bmatrix}0 & 1 \\1 & 0 \end{bmatrix}$$

we find a matrix multiplication table as below

$$\begin{array}{|c|c|c|c|c|}

\hline & M\left( e \right) & M\left( a \right) & M\left( b \right) & M\left( c \right) \\

\hline M\left( e \right) & M\left( e \right) & M\left( a \right) & M\left( b \right) & M\left( c \right) \\

\hline M\left( a \right) & M\left( a \right) & -M\left( e \right) & M\left( c \right) & - M\left( b \right) \\

\hline M\left( b \right) & M\left( b \right) & - M\left( c \right) & M\left( e \right) & -M\left( a \right) \\

\hline M\left( c \right) & M\left( c \right) & M\left( b \right) & M\left( a \right) & M\left( e \right) \\ \hline \end{array}

$$

This shows that the matrices we are now using form a representation up to a phase, or a projective representation.

 

[rocdoc]

Been thinking, just a little, not a lot. There seem to be much simpler examples.