How does GAP understand $SL_2(\mathbb{F}_3)$?

Question

• When one is asking for " IrreducibleRepresentations(SL(2,3))" it makes sense that the matrices it returns are over the complex field as thats the field on which the representations are. There GAP uses something called E(3) which I believe is a primitive cuberoot of unity.

  But when one is asking for "List(GeneratorsOfGroup(SL(2,3)));" what is this Z(3) that GAP is using in the entries of the two matrices that it outputs? (it outputs [[0,1],[Z(3),0]] and [ [ 0, 1 ], [ Z(3), 0] ])

When one is listing out the generators of SL(2,3) then shouldn't the matrices be 2 dimensional matrices with entries in F_3 ?

• How does one know what is the "action isomorphism" that GAP is using to label the generators of SL(2,3) by permutations? As of now I have no clue as to which of the 24 elements of $SL_2(\mathbb{F}_3)$ did GAP denote as "(4,5,6)(7,9,8)" and "(2,7,3,4)(5,8,9,6)" and how was this obtained!

  Is GAP choosing an ordering of the 24 elements of SL(2,3) and hence the permutation representation of a group element corresponds to the permutation induced on this ordered list by the action of this element? - If yes then to understand what is the meaning of "(4,5,6)(7,9,8)" and "(2,7,3,4)(5,8,9,6)" - the notation that GAP uses to denote its two generators of SL(2,3) - one needs to know what is the indexing of the elements of the group used by GAP internally!

• Is there a way I can get GAP to give me the word decomposition of all the 24 elements in this group in terms of the two matrices that it uses as generators? - the two matrices returned as output to "List(GeneratorsOfGroup(SL(2,3)));"

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I think there are two natural ways to think of F_p,

• as the set {0,1,2,3..,p-1}

• or as the set {0,1,x,x^2,...,x^{p-2}}

where x is a (p-1)^th primitive root of unity

(..and I believe the above goes through even if p were to be some non-trivial power of a prime number..)

Now what is GAP thinking when it says that [[0,1],[Z(3),0]] and [ [ 0, 1 ], [ Z(3), 0] ] are its generators of SL_2(F_3)?

Two natural ways to represent F_3 can either be over {0,1,-1} or over {0,1,2}

Answer 1

Z(3) is a (particular chosen) generator for the multiplicative group (i.e. the $x$ in your second model). You might want to read the Wikipedia articles on http://en.wikipedia.org/wiki/Zech%27s_logarithm and http://en.wikipedia.org/wiki/Conway_polynomial_(finite_fields) to understand why this is chosen.

The display of 1 and 0 is purely as a convenience, in fact (if you use `Print' you see this) the first generator is [[0*Z(3),Z(3)^0],[Z(3),0*Z(3)]] which is a matrix over $\mathbb{F}_3$.

The action homomorphism used is on the (nonzero, row) vectors. You can get this list (including the zero vector) for example as Elements(GF(3)^2).

The manual section http://www.gap-system.org/Manuals/doc/ref/chap39.html#X7E19F92284F6684E (Expressing Group Elements as Words in Generators) describes how to express elements as words.