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Permutation Representations and the Modular Group

asked 2012-08-03 11:19:42 +0200

Jimeree gravatar image


Given a congruence subgroup of the modular group G = SL2Z, how can one use Sage to find the permutation representations of G on the cosets of each of those normal subgroups? Specifically I am looking at the subgroups on page 22 here:

Many thanks!

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answered 2012-08-03 23:35:58 +0200

vdelecroix gravatar image


Given a congruence subgroup, you can get another data structure for it given exactly by the action of the standard generators (L = parabolic fixing infinity, R = parbolic fixing 0, S2 = element of order 2, S3 = element of order 3). You may proceed as follows for the first example of the list

sage: G = Gamma(4)
sage: GG = G.as_permutation_group()  
sage: GG  
Arithmetic subgroup of index 48
sage: GG.L() 
sage: GG.R()
sage: GG.S2()
sage: GG.S3()

Warning: it is the right action on right coset (ie of the form Hg).

Note that you can also plot a fundamental domain using Farey symbols (Kulkarni method)

sage: G = Gamma(4)
sage: FareySymbol(G).fundamental_domain()


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Thanks very much! And I have a follow-up question too: Is there an easy way to find the permutation representations of PSL(2,Z) (rather than SL(2,Z)) on the cosets of these congruence subgroups? I notice there is a projective_index() command, so is there anything similar which could do the job here? Many thanks,

Jimeree gravatar imageJimeree ( 2012-08-06 05:34:42 +0200 )edit

Actually, if -Id is in your subgroup (ie it is an even subgroup) the choice of PSL(2,Z) or SL(2,Z) does not matter. When you have an odd subgroup, you need to add -Id to your subgroup which can be done with the method .to_even_subgroup() available for subgroup of SL(2,Z) represented by permutation.

vdelecroix gravatar imagevdelecroix ( 2012-08-06 07:40:35 +0200 )edit

Great! Thanks very much

Jimeree gravatar imageJimeree ( 2012-08-06 09:49:52 +0200 )edit

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Asked: 2012-08-03 11:19:42 +0200

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Last updated: Aug 03 '12