Choosing Particular representation and/or identifying the representation of the group in GAP

Question

I have three questions regarding the usage of GAP:

1. I want to choose a particular matrix representation of a group, and thanks to the previous answer, I can display which representations are available, i.e., DisplayAtlasInfo("2.A5", Characteristic, 0); gives me the following answer:

   19: G <= GL(8b, Z)

   20: G <= GL(12, Z)

   21: G <= GL(2a, Field([E(5)]))

   22: G <= GL(4a, Field([Sqrt(-1)]))

   23: G <= GL(4a, Field([Sqrt(-6)]))

   24: G <= GL(6, Field([Sqrt(-1)]))

   25: G <= GL(6, Field([Sqrt(-3)]))

   26: G <= GL(1a, QuaternionAlgebra([Sqrt(5)]))

I want to choose this representation: GL(1a, QuaternionAlgebra([Sqrt(5)])). I thought that I could write the following script:

G := GL(1a, QuaternionAlgebra([Sqrt(5)]))

However, I got the answer saying Variable:

'1a' must have a value not in any function at stdin:16.

When I remove 1a, then I got the error:

<function "GeneralLinearGroup">( ) called from read-eval loop at qua2.g:2

. I do not know what 1a means. (I have also tried possible combinations, i.e., writing 2.A5 instead of G or writing SL(2,5) instead of GL, writing 1a in the "" etc...) How do I choose the particular matrix representation in GAP?

2. My second question is, I can find the description of the group by writing StructureDescription(G) where G is the group I created. However, as I recently noticed, there are many possible ways to write the same group, and I am wondering if it is possible to find which representation I am finding, i.e., does Representation(G) return me the Atlas Group or quaternion algebra or complex irreducible representation? Is it possible, or am I asking for too much from GAP? P.S I tried ?FamilyDescription and ?RepresentationDescription and ?MatrixDescription to see if manual has something but could not see anything for now

3. Is it also possible to check if two given matrix groups are equal or not with a function? I can, of course, do that simply by comparing the elements of the groups by iterating through them with for loops, but depending on the order of the group, it can take a long time. So, I am wondering if there is an "equal" like function in GAP that can handle this more efficiently? (for example, I can create a group simply by defining the generator matrices by hand and I can also do that directly using GAP functionality. I produce group with the same orders but the matrix elements could be different. ) I am aware that I am asking so many questions and potentially bothering people all the time. I apologize for the interruption and thank you for your patience

Answer 1

1. I think you misunderstand the output of DisplayAtlas Info. The letters (a,b) are there to distinguish inequivalent representations. And the images are not equal(as you assign) but only subgroups.

2. The expression "the description of the group" makes no sense.You can ask for "one description", but there is no scheme that guarantees that descriptions identify groups up to isomorphism.

3. You can compare groups and matrices with the = operator. However underlying this (and previous) questions seems to be a misunderstanding between equality of groups and the two equivalence relations given by GL-conjugacy or isomorphism. I think you want to construct one particular group but you use concepts that are only defined up to equivalence.