Cayley graph of PGL(2,q)

[Oren Becker]

I would like to compute the spectrum of a certain Cayley graph on PGL(2,q), for a certain prime power q.

I created the generators of the graph as elements of GL(2,q).

Now, how do I get the spectrum of the Cayley graph on PGL(2,q)? I thought to first create the Cayley graph using PGL.cayley_graph(), and then take the spectrum. But, taking a quotient is not implemented on GL(2,q), so I can't even move from GL to PGL.

Of course, I can work hard to create the adjacency matrix, then take the spectrum, but I'm hoping for an easier solution.

(I am new to Sage).

Thanks,
Oren.

[David Joyner]

On Mon, Apr 7, 2014 at 11:36 AM, Oren Becker <oren....@gmail.com> wrote:
> I would like to compute the spectrum of a certain Cayley graph on PGL(2,q), for a certain prime power q.
>
> I created the generators of the graph as elements of GL(2,q).
>

Why don't you redo your steps but change this step to use PGL instead of GL?

> Now, how do I get the spectrum of the Cayley graph on PGL(2,q)? I thought to first create the Cayley graph using PGL.cayley_graph(), and then take the spectrum. But, taking a quotient is not implemented on GL(2,q), so I can't even move from GL to PGL.
>
> Of course, I can work hard to create the adjacency matrix, then take the spectrum, but I'm hoping for an easier solution.
>
> (I am new to Sage).
>
> Thanks,
> Oren.
>

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[Oren Becker]

My steps created matrices. The group PGL is implemented as a group of permutations.

I can make my question more accurate:

I have a (symmetric) set of 2x2 matrices S over Fq. How can I find the spectrum of the Cayley graph of PGL(2,q) with respect to the set of generators S?

[Dima Pasechnik]

On 2014-04-07, Oren Becker <oren....@gmail.com> wrote:
> My steps created matrices. The group PGL is implemented as a group of
> permutations.
>
> I can make my question more accurate:
>
> I have a (symmetric) set of 2x2 matrices S over Fq. How can I find the
> spectrum of the Cayley graph of PGL(2,q) with respect to the set of
> generators S?

this is easy to do in GAP using the GRAPE package (part of
gap_packages spkg)

Use GAP's functionality to convert GL into PGL (you can make GL act on
lines, as a permutation group), then call CayleyGraph.


(you then can import the resulting graph into Sage using the interface
with GAP and compute the eigenvalues)

Should you need more details to implement this plan,
please ask...

HTH,
Dmitrii