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It is well known that PSL(2,7) contains 168 elements. I'm looking for a method of obtaining irreducible representations (the matrices, not just the character table) which as implied by the title are: 1) over the real or complex fields 2) faithful, i.e. isomorphic to PSL(2,7)

I'd prefer an analytic expression for such matrices if this is possible.

Can the question be answered for the more general case of PSL(2,p), p prime, or even the more ambitious PSL(n,p)?

Thanks!

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    $\begingroup$ In GAP, the command is IrreducibleRepresentations(PSL(2,7)); $\endgroup$ – John Wiltshire-Gordon Oct 26 '12 at 22:21
  • $\begingroup$ @John Wiltshire-Gordon what about irreps over reals ? $\endgroup$ – Alexander Chervov Oct 27 '12 at 12:33
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The irreducible character degrees for this group have degree 1,3,3,6,7,8. To get the irreducible degree $7$-representation, induce a non trivial linear character of either of the maximal parabolics isomorphic to $S_4$. To get the degree $8$ irreducible, induce a non-trivial linear character of the Sylow $7$-normalizer ( a Frobenius group of order $21$). To get the degree $6$ irreducible, induce the trivial character of ether of the parabolics isomorphic to $S_{4},$ obtaining an orthogonal representation. Then take the orthogonal complement of the $1$-dimensional fixed-point space. To get one of the two $3$-dimensional representations, induce a non-trivial linear character of the Sylow $7$-subgroup. The already constructed (unitary) $8$-dimensional representation shows up, as does the $6$-dimensional (unitary) repesentation and the $7$-dimensional unitary irreducible representation. Take the orthogonal complement of the sum of these. This gives a $3$-dimensional unitary representation. Take the dual of that as well, and you have all (non-trivial) irreducible representations, up to equivalence.

Later edit: Note that for the irreducible characters of degree $6,7$ and $8$ the representations above may be explicitly given as real representations. The degree $8$ representation requires a little further thought. If we induce the trivial character from a Sylow $2$-subgroup, the $8$ dimensional irreducible character occurs with multiplicity $1,$ the trivial character occurs once, the $7$-dimensional irreducible character does not occur, the $6$-dimensional character occurs once and the two three dimensional character each occur once. Since the permuation representation is realized over $\mathbb{R},$ it follows that the $8$-dimensional representation may be reaized over the real field.

The two $3$-dimensional representations do not have real characters. To obtain a real representation of their sum, do the last procedure instead with a real irreducible two dimensional orthogonal representation of the Sylow $7$-subgroup (which is just as group of real rotations): two copies of each of the $6,7$ and $8$ degree real orthogonal representations show up, so take the orthogonal complement of their sum, obtaining an orthogonal representation of degree $6$ which is irreducible as a real representation, but not a complex representation.

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  • $\begingroup$ ${\rm PSL}(2,7)$ is one of the exceptional ${\rm PSL}(2,p)$'s which has a transitive permutation of degree $p$, and this is exploited in the above explanation. This does not happen for $p>11$ as Galois already knew. Frobenius knew the character table of ${\rm PSL}(2,p),$ and as Jim Humphreys say, it is possible to deduce explicit irreducible complex representations to match the characters, by decomposing various induced representations $\endgroup$ – Geoff Robinson Oct 27 '12 at 14:03
  • $\begingroup$ sorry what means " transitive permutation of degree p" ? $\endgroup$ – Alexander Chervov Oct 27 '12 at 18:13
  • $\begingroup$ can similar ideas be used for other groups gl/psl(n,q) ? $\endgroup$ – Alexander Chervov Oct 27 '12 at 18:17
  • $\begingroup$ Transitive permutation group of degree $p$ for $p$ prime just means transitive subgroup of the symmetric group $S_{p}.$ For the second question, refer to the answer by Jim Humphreys. $\endgroup$ – Geoff Robinson Oct 27 '12 at 18:20
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    $\begingroup$ One way is to see hat as it is also ${\rm GL}(3,2)$ it has a transitive permutation representation on the non-zero vevtors in a $3$-dimensional vector space over the field of $2$-elemeents. Another was is to not hat it has subgroups of index $7$ (in fact, two conjugacy classes of subgroups, each isomorphic to $S_{4}).$ $\endgroup$ – Geoff Robinson Oct 27 '12 at 23:17
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For a single group of relatively small order, computer methods are available, as pointed out by John Wiltshire-Gordon. For a general theoretical approach, there is much less to go on in terms of methods. But in my old expository article in Amer. Math. Monthly 82 (1975), I did reference the work done (before the Deligne-Lusztig era) on representations of finite special linear groups over finite fields. Besides the character theory worked out by Frobenius (and the American H. Jordan independently), there are only a couple of serious attempts to describe the actual representations. The two references I could find are:

S. Tanaka, Construction and classification of irreducible representations of the special linear group of the second order over a finite field, Osaka J. Math., 4 (1967) 65-84

S. I. Gel'fand, Representations of the full linear group over a finite field, Math. USSR-Sb., 12 (1970) 13-39

Neither of these seems to have given much insight into more complicated finite groups of Lie type, but taken on their own they are worth looking at. I'm not sure what is currently available online, but the second article appears in a fairly standard translation journal. Beyond the simple groups of rank 1, it seems quite challenging to say anything concrete about the matrix description of arbitrary irreducible representations. Though there is the underlying Harish-Chandra philosophy for Lie groups which certainly replicates here at least in the Deligne-Lusztig character theory.

Some added comments:

1) The paper by Tanaka is freely available through the Euclid Project here. This is closest to the question asked about $\mathrm{SL}_2(\mathbb{F}_p)$, but like Gelfand's work is fairly uniform over any finite field. To pass to the projective group is usually straightforward, by taking into account which representations are trivial at the central element $-I$.

2) At the 1971 Budapest summer school, Sergei Gelfand gave an exposition (in English) of his work, published in the proceedings some years later: Representations of the general linear group over a finite field. Lie groups and their representations (Proc. Summer School on Group Representations of the Bolyai Janos Math. Soc., Budapest, 1971), pp. 119–132. Halsted, New York, 1975.

3) While it's easy to pass from a special linear group to its simple quotient, it's much trickier to pass from a general linear to a special linear subgroup (even on the level of characters), so Gelfand's approach may not be so directly useful here. The concrete work in rank 1 also seems to be of little help in higher ranks but has been exploited in work on representations over local fields.

4) In the rank 1 case, roughly half of the irreducible representations (or characters) are easy to construct using induction from a Borel subgroup; even for finite groups, a functional description of induction is traditional here. But the big problem is to pin down the cuspidal or discrete series representations (called "analytic" by Gelfand). Both Tanaka and Gelfand use essentially analytic approaches to the problem, via notions such as "Bessel function" over a finite field.

5) To get real representations, other people have suggested approaches. The well-known character table provides some guidance here.

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There is much interesting information in the following article :

Noam D. Elkies, The Klein Quartic in Number Theory, The eightfold way, 51–101, Math. Sci. Res. Inst. Publ., 35, Cambridge Univ. Press, Cambridge, 1999.

See in particular Section 1.3 for questions about the fields of definition of the representations.

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Incidentally I was interested in this question a couple of months ago, may be these remarks will be useful.

1) PSL(2,7) remarkably isomorphic (MO 37525) to GL(3,2). (See also Wikipedia page and Vipul Naik's site).

2) It is simple group so all irreps are faithful.

3) There are several on-line expositions around irreps of GL(2,F_q), SL(2,F_q). Not sure PSL(2,F_q) is there, but nevertheless might be useful to give these links:

Paul Garrett: http://www.math.umn.edu/~garrett/m/v/toy_GL2.pdf

Etingof&Students: http://arxiv.org/abs/0901.0827

Amritanshu Prasad http://www.imsc.res.in/~amri/html_notes/notes.html#notesch2.html

(This also discusses GL(n,F_q) partly).

Also the book by Fulton Harris discusses the GL(2,F_q) irreps.

4) The group PSL(2,F_q) has irrep of dimension "q" which actually can realized over reals (actually rationals). It can be seen like this: PSL(2,F_q) acts on the projective line P^1(q). This is a set of q+1 points. So we have a q+1 representation realized in functions on this set - as any permutation representation it contains the constants as 1d sub-irrep. The complement to constants will give you q dim. irrep.

6) For the particular case of PSL(2,7) we may try to construct irreps inducing from the Sylow's cyclic subgroups of orders 7 and 3. You might enjoy M.Isaacs MO answer on my question about decomposition of these irreps.

7) These were over mainly about complex numbers irreps. I cannot say much about irreps over reals, but here are some remarks. There are 6 irreps over complex numbers (I have inserted character table to Wikipedia page) dimensions 1, 3,3, 6,7,8. Actually 1,6,7,8 can be realized over reals. About 7 - see discussion above, 1d - is trivial about 6,8 - let me refer to this page which calculates Frobenius-Schur indicators and shows that they are equal to 1 (hence irreps realized over reals) for 1,6,7,8.

Now there are two 3d irreps (cuspidal irreps) they are complex conjugate to each other, so their direct sum can be realized over reals.

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