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May
2
revised How does GAP understand $SL_2(\mathbb{F}_3)$?
added 23 characters in body
May
2
revised How does GAP understand $SL_2(\mathbb{F}_3)$?
added 44 characters in body
May
2
revised How does GAP understand $SL_2(\mathbb{F}_3)$?
added 107 characters in body
May
2
comment How to read GAP's output on “IrreducibleRepresentations”?
Can you kindly give the link to where in the manual this is explained? In the manual I can't even find an index to see what is the meaning of Z(3) !
May
2
asked How does GAP understand $SL_2(\mathbb{F}_3)$?
May
2
comment How to read GAP's output on “IrreducibleRepresentations”?
@Max Its not that clear. When one is asking for " IrreducibleRepresentations(SL(2,3))" it makes sense that the matrices it returns are over the complex field as thats where the representations are. Then GAP uses something called E(3) which I believe is the primitive cuberoot of unity. But when one is asking for List(GeneratorsOfGroup(SL(2,3))); what is this Z(3) that GAP is using? When one is listing out the generators of SL(2,3) the matrices should be 2x2 matrices with entries in F_3.
May
1
comment How to read GAP's output on “IrreducibleRepresentations”?
Do you understand what I am trying to say? Its not clear as to in which representation of F_3 is it possible for 0, 1 and a primitive root of unity to occur together! Why is GAP shifting from the Z(3) that it uses in the generators to the E(3) that it uses later?
May
1
comment How to read GAP's output on “IrreducibleRepresentations”?
This interpretation of E(3) or Z(3) doesn't make sense. Firstly is E(3) the same as Z(3) ? 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 the (p-1)^th primitive root of unity Now what is GAP thinking when it says that [[0,1],[Z(3),0]] is one its generators of SL_2(F_3)? F_3 can either be represented in {0,1,-1} or as {0,1,2} What is Z(3) or E(3) ?
May
1
comment How to read GAP's output on “IrreducibleRepresentations”?
This is not clear. What is this E(3) ? If 0 is occurring in the generators that GAP is using then it is having F_3 in the additive representation which has elements (0,1,-1) and there is no clear meaning to E(3). What is permutation representation that GAP is using to write its generators in a cycle decomposed notation?
Apr
30
comment What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?
For q a prime power, isn't the permutation action of GL_2(F_q) or SL_2(F_q) on F_q^2 giving a q^2 dimensional representation? I would like to break this into irreps over C - obviously.
Apr
30
comment What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?
What is C(X)? Given the permutation action of G on X which is the |X| sized representation here?
Apr
30
comment What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?
(1) Both GL_2(F_q) and SL_2(F_q) have a (q+1)-dimensional representation on P(F_q^2) - agreed? (2) There is no irrep of GL_2(F_q) or SL_2(F_q) which is of dimension q^2 (as this permutation representation is) then how can this be irreducible? (you can see the irrep sizes here, groupprops.subwiki.org/wiki/…, groupprops.subwiki.org/wiki/…
Apr
30
comment What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?
What we have defined here are group actions for SL_2(F_p), PSL_2(F_p), GL_2(F_p) and PGL_2(F_p) on the vector space F_p^2? Right? Are any of these irreducible? If not what is their irrep decomposition?
Apr
30
accepted What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?
Apr
30
comment What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?
I am not understanding what you are not understanding :D Here is an action of these groups on $\mathbb{F}_p^2$ (or on $\mathbb{P}(\mathbb{F}_p^2 )$) so we have a representation of these groups on these vector spaces. (so we have here basically listed out 6 representations) Are any of these irreducible? If not then what is the irrep decomposition of these representations?
Apr
30
comment What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?
@Qiachu Yuan What I need is an enumeration of all the group elements of these groups, SL_2(F_q) or PSL_2(F_q) or GL_2(F_q) or PGL_2(F_q), so that I can do these calculations by hand for specific cases. Is this somehow available?
Apr
30
comment What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?
@Qiachu Yuan Thanks for the help! Do you know what the irrep decomposition is of these permutation representations?
Apr
30
comment What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?
@QiaochuYuan See my comments to Herbert's answer.
Apr
30
comment What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?
And equivalently is there a way to explicitly enumerate all the elements of any of these groups?
Apr
30
comment What is the permutation representation of $SL_2(\mathbb{F}_p)$, $PSL_2(\mathbb{F}_p)$ and $GL_2(\mathbb{F}_p)$?
Yes, obviously. But can you give me the action? As in given a $2 \times 2$ matrix over $\mathbb{F}_p$ an element of say $SL_2(F_p)$ what is its action on some element say $(a,b) \in \mathbb{F}_p^2$ ?