> < ^ Date: Fri, 17 May 2002 17:04:44 +0200
> < ^ From: Thomas Breuer <Thomas.Breuer@Math.RWTH-Aachen.DE >
< ^ Subject: Re: pgu

Dear GAP Forum,

Luca Giuzzi wrote

I need to be able to obtain the action of the Projective Unitary
group PGU(n+1,q) on the vectors of the Projective Space PG(n,q^2).
[actually, I need this for q=5 and n=3, but I expect the solution
to be valid for any suitable n,q]
Here there is my question:
What is the sesquilinear form respected by the group PGU(n+1,q),
assuming that the vectors of PG(3,q^2) are enumerated in the order
given by
Orbit(GL(4,q^2),Z(q)^0*[1,0,0,0],OnLines) ?
[or, conversely, in what order may I enumerate the elements of
PG(n,q^2) if I wish PGU(n+1,q) to be the group preserving the
sesquilinear form induced by the identity matrix?]

Perhaps the most natural thing to start with is the matrix group GU(4,q),
which has the invariant form stored via a matrix.

```gap> n:= 3;;  q:= 5;;
gap> g:= GU(n+1,q);
GU(4,5)
gap> form:= InvariantSesquilinearForm( g );
rec(
matrix := [ [ 0*Z(5), 0*Z(5), 0*Z(5), Z(5)^0 ], [ 0*Z(5), 0*Z(5), Z(5)^0,
0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ],
[ Z(5)^0, 0*Z(5), 0*Z(5), 0*Z(5) ] ] )
gap> Display( form.matrix );
. . . 1
. . 1 .
. 1 . .
1 . . .
```

Now one can consider the projective action on the normed vectors.
Each element of GU(4,q) induces a permutation which can be computed
via `Permutation', with the action specified by `OnLines';
the group of these permutations is isomorphic with PGU(4,q).

```gap> vectors:= Orbit(GL(4,q^2),Z(q)^0*[1,0,0,0],OnLines);;
gap> g.1;
[ [ Z(5^2), 0*Z(5), 0*Z(5), 0*Z(5) ],
[ 0*Z(5), Z(5)^0, 0*Z(5), 0*Z(5) ],
[ 0*Z(5), 0*Z(5), Z(5)^0, 0*Z(5) ],
[ 0*Z(5), 0*Z(5), 0*Z(5), Z(5^2)^19 ] ]
gap> pi:= Permutation( g.1, vectors, OnLines );
(2,62,2957,15647)(3,122,5114,16171)(4,16137,8589,11189,33,853,16092,10633,
459,911,15130,11744,5115,4996,12332,448,14459,5061,121,11770,15838,14920,
8047,7386)(5,239,8210,16251)(6,2865,5010,11720,63,184,11739,13435,876,1662,
12758,7617,8211,2869,14831,14757,12947,15340,15660,14022,11799,10137,16083,
5082)(7,15375,8504,14336,64,1590,16203,13960,877,1693,16054,14658,8212,
8065,11612,856,10697,8145,237,14675,12884,15997,11710,10991)(8,3228,1700,
1677,95,5126,177,609,891,617,5459,4542,1860,11925,170,3480,1232,144,3587,
114,6589,13785,5575,495)(9,458,11895,16265)(10,12782,14748,13646,123,296,
16021,14720,1628,2979,6838,15468,11896,8150,15776,14092,5003,15167,11438,
10598,1592,14548,12484,15645)(11,4987,8084,11492,124,351,14232,15504,1629,
3003,15189,11239,11897,4993,15814,15268,13558,14666,15704,12670,14697,
13576,8057,8169)(12,3237,1704,1687,154,8401,14042,633,1821,728,3956,4853,
1900,9296,201,5704,8097,148,3184,475,8334,12823,5972,502)(13,16114,12156,
15823,126,2892,10530,11544,1631,3050,15693,14000,11898,11730,14588,1596,
14001,11819,454,8041,6536,445,14335,14204)( [...] )
```

Kind regards,
Thomas

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