Is there a review/exposition of the representation theory of $PSL_2(\mathbb{F}_q)$ ? Like an enumeration of its irreducible representations and their dimensions as a function of $q$?
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2$\begingroup$ It's complex character table is well known, I think it was known to Frobenius. $\endgroup$– Geoff RobinsonCommented Apr 18, 2015 at 22:32
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3 Answers
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Jeff Adams has comprehensive notes on his website: http://www.math.umd.edu/~jda/characters/characters.pdf
The irreducible characters of the groups SL(2), PGL(2), GL(2) and PSL(2) over finite fields are described.
The problem for PSL(2) was originally solved by Frobenius in his 1896 paper: Über Gruppencharaktere. It is the first article in Volume 3 of his collected works (Gesammelte Abhandunglen, Band III). On the last page of this article, I found the table:
answered Apr 19, 2015 at 2:33
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$\begingroup$ The comment on the last line of Frobenius's paper is nice: $\mathrm{PSL}(2, F_3)\cong A_4$ and $\mathrm{PSL}(2, F_5)\cong A_5$. $\endgroup$ Commented Apr 19, 2015 at 10:09
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$\begingroup$ Thanks for the help! On page 12 can you kindly explain how to read it that table? What is the "dimension", "size", "number", "number" ? Can you kindly clarify the notation? $\endgroup$– user6818Commented Apr 19, 2015 at 21:50
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$\begingroup$ The number of conjugacy classes depends on $q$, so you can't really list them all separately. Fortunately, it is possible to group the classes into types; all the classes of the same type have the same size. This is what "size" and "number" refer to for the classes - if number is n and size is s, then there are n classes of size s. For representations again a similar principle holds. Both classes and representations of a given type have parameters which appear in the character formulas. To get the hang of the notation used in the parameters, you have to read through the earlier sections. $\endgroup$ Commented Apr 20, 2015 at 5:19
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$\begingroup$ Thanks! Let me try to decipher this! BTW, what I am looking for is this : I am looking for a sequence of groups of increasing size such that they all have a representation in terms of permutation matrices and the dimension of the smallest non-trivial irreducible representation in these permutation representations is as large as possible. Any help? $\endgroup$– user6818Commented Apr 20, 2015 at 15:31
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$\begingroup$ In this case, the group $PSL(2, q)$ has order $q(q^2-1)/2$. A representation of dimension $q+1$ is an irreducible representation which occurs in a permutation representation (the representation induced from a trivial representation of the subgroup of upper triangular matrices). $\endgroup$ Commented Apr 21, 2015 at 3:56
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How about the MAA article of J. E. Humphreys?
answered Apr 18, 2015 at 22:59
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1$\begingroup$ @FanZheng I believe following the link would answer your question $\endgroup$ Commented Apr 18, 2015 at 23:32
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2$\begingroup$ Maybe the text before Humphrey's article is worth reading and pondering, isnt'it ? $\endgroup$– BS.Commented Apr 18, 2015 at 23:51
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Cedric Bonnafe wrote a book entitled "Representatons of $\text{SL}_2(\mathbb{F}_q)$" that has evertyhing you are looking for.
answered Apr 19, 2015 at 4:23