GAP (FinInG) - Appendix B: The finite classical groups in FinInG

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B The finite classical groups in FinInG
B.1 Standard forms used to produce the finite classical groups.
B.1-1 CanonicalGramMatrix
B.1-2 CanonicalQuadraticForm
B.2 Direct commands to construct the projective classical groups in FinInG
B.2-1 SOdesargues
B.2-2 GOdesargues
B.2-3 SUdesargues
B.2-4 GUdesargues
B.2-5 Spdesargues
B.2-6 GeneralSymplecticGroup
B.2-7 GSpdesargues
B.2-8 GammaSp
B.2-9 DeltaOminus
B.2-10 DeltaOplus
B.2-11 GammaOminus
B.2-12 GammaO
B.2-13 GammaOplus
B.2-14 GammaU
B.2-15 G2fining
B.2-16 3D4fining
B.3 Basis of the collineation groups
B.3-1 FindBasePointCandidates

### B The finite classical groups in FinInG

#### B.1 Standard forms used to produce the finite classical groups.

An overview of operations is given that produce gram matrices to construct standard forms. The notion standard form is explained in Section 7.2, in the context of canonical and standard polar spaces.

##### B.1-1 CanonicalGramMatrix

`‣ CanonicalGramMatrix`( type, d, f ) | ( operation )

Returns: a Gram matrix usable as input to construct a sesquilinear form

The arguments d and f are the vector dimension and the finite field respectively. The argument type is either "symplectic", "hermitian", "hyperbolic", "elliptic" or "parabolic".

If type equals "symplectic", the Gram matrix is

0 | 1 | 0 | 0 | ... | 0 | 0
-1 | 0 | 0 | 0 | ... | 0 | 0
0 | 0 | 0 | 1 | ... | 0 | 0
0 | 0 | -1 | 0 | ... | 0 | 0
: | : | : | : | ··· | : | :
0 | 0 | 0 | 0 | ... | 0 | 1
0 | 0 | 0 | 0 | ... | -1 | 0

If type equals "hermitian", the Gram matrix is the identity matrix of dimension d over the field f=GF(q)

If type equals "hyperbolic", the Gram matrix is

0 | a | 0 | 0 | ... | 0 | 0
a | 0 | 0 | 0 | ... | 0 | 0
0 | 0 | 0 | a | ... | 0 | 0
0 | 0 | a | 0 | ... | 0 | 0
: | : | : | : | ··· | : | :
0 | 0 | 0 | 0 | ... | 0 | a
0 | 0 | 0 | 0 | ... | a | 0

with a=(p+1)/2 if p+1 = 0 mod 4, q=ph and a=1 otherwise.

If type equals "elliptic", the Gram matrix is

1 | 0 | 0 | 0 | ... | 0 | 0
0 | t | 0 | 0 | ... | 0 | 0
0 | 0 | 0 | a | ... | 0 | 0
0 | 0 | a | 0 | ... | 0 | 0
: | : | : | : | ··· | : | :
0 | 0 | 0 | 0 | ... | 0 | a
0 | 0 | 0 | 0 | ... | a | 0

with t the primitive root of GF(q) if q = 1 mod 4 or q = 2 mod 4 , and t=1 otherwise; and a=(p+1)/2 if p+1 = 0 mod 4, q=ph and a=1 otherwise.

If type equals "parabolic", the Gram matrix is

t | 0 | 0 | ... | 0 | 0
0 | 0 | a | ... | 0 | 0
0 | a | 0 | ... | 0 | 0
: | : | : | ··· | : | :
0 | 0 | 0 | ... | 0 | a
0 | 0 | 0 | ... | a | 0

with t the primitive root of GF(p) and a=t(p+1)/2 if q = 5 mod 8 or q = 7 mod 8 , and t=a=1 otherwise.

There is no error message when asking for a hyperbolic, elliptic or parabolic type if the characteristic of the field f is even. In such a case, a matrix is returned, which is of course not suitable to create a bilinear form that corresponds with an orthogonal polar space. For this reason, `CanonicalGramMatrix` is not a operation designed for the user.

##### B.1-2 CanonicalQuadraticForm

`‣ CanonicalQuadraticForm`( type, d, f ) | ( operation )

Returns: a Gram matrix usable as input to construct a quadratic form

The arguments d and f are the vector dimension and the finite field respectively. The argument type is either "hyperbolic", "elliptic" or "parabolic". The matrix returned can be used to construct a quadratic form.

If type equals "hyperbolic", the Gram matrix returned will result in the quadratic form x1x2+...+ xd-1xd

If type equals "elliptic", the Gram matrix returned will result in the quadratic form x12+x1x2+νx22 ...+ xd-1xdwith ν=αi, with α the primitive element of the multiplicative group of GF(q), which is in GAP `Z(q)`, and i the first number in [0,1,...,q-2] for which x2+x+ν is irreducible over GF(q).

If type equals "parabolic", the Gram matrix returned will result in the quadratic form x12+x2x3+... + xd-1xd

This function is intended to be used only when the characteristic of f is two, but there is no error message is this is not the case. For this reason, `CanonicalQuadraticForm` is not an operation designed for the user.

#### B.2 Direct commands to construct the projective classical groups in FinInG

As explained in Chapter 7, Section 7.7, we have assumed that the user asks for the projective classical groups in an indirect way, i.e. as a (subgroup) of the collineation group of a classical polar space. However, shortcuts to these groups exist. More information on the notations can be found in Section 7.7.

##### B.2-1 SOdesargues

`‣ SOdesargues`( e, d, f ) | ( operation )

Returns: the special isometry group of a canonical orthogonal polar space

The argument e determines the type of the orthogonal polar space, i.e. -1,0,1 for an elliptic, hyperbolic, parabolic orthogonal space, respectively. The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `SO`, a GAP command returning the appropriate matrix group. Internally, the invariant form is asked, and the base change to our canonical form is obtained using the package form

    gap> SOdesargues(-1,6,GF(9));
    PSO(-1,6,9)
    gap> SOdesargues(0,7,GF(11));
    PSO(0,7,11)
    gap> SOdesargues(1,8,GF(16));
    PSO(1,8,16)

##### B.2-2 GOdesargues

`‣ GOdesargues`( e, d, f ) | ( operation )

Returns: the isometry group of a canonical orthogonal polar space

The argument e determines the type of the orthogonal polar space, i.e. -1,0,1 for an elliptic, hyperbolic, parabolic orthogonal space, respectively. The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `GO`, a GAP command returning the appropriate matrix group. Internally, the invariant form is asked, and the base change to our canonical form is obtained using the package form

    gap> GOdesargues(-1,6,GF(9));
    PGO(-1,6,9)
    gap> GOdesargues(0,7,GF(11));
    PGO(0,7,11)
    gap> GOdesargues(1,8,GF(16));
    PGO(1,8,16)

##### B.2-3 SUdesargues

`‣ SUdesargues`( d, f ) | ( operation )

Returns: the special isometry group of a canonical hermitian polar space

The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `SU`, a GAP command returning the appropriate matrix group. Internally, the invariant form is asked, and the base change to our canonical form is obtained using the package form

    gap> SUdesargues(4,GF(9));
    PSU(4,3^2)

##### B.2-4 GUdesargues

`‣ GUdesargues`( d, f ) | ( operation )

Returns: the isometry/similarity group of a canonical hermitian polar space

The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `GU`, a GAP command returning the appropriate matrix group. Internally, the invariant form is asked, and the base change to our canonical form is obtained using the package form

    gap> GUdesargues(4,GF(9));
    PGU(4,3^2)

##### B.2-5 Spdesargues

`‣ Spdesargues`( d, f ) | ( operation )

Returns: the (special) isometry group of a canonical symplectic polar space

The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `Sp`, a GAP command returning the appropriate matrix group. Internally, the invariant form is asked, and the base change to our canonical form is obtained using the package form

    gap> Spdesargues(6,GF(11));
    PSp(6,11)

##### B.2-6 GeneralSymplecticGroup

`‣ GeneralSymplecticGroup`( d, f ) | ( operation )

Returns: the isometry group of a canonical symplectic form

The argument d is the dimension of the underlying vector space, f is the finite field. Internally, the invariant form is asked, and the base change to our canonical form is obtained using the package form

    gap> GeneralSymplecticGroup(6,GF(7));
    GSp(6,7)

##### B.2-7 GSpdesargues

`‣ GSpdesargues`( d, f ) | ( operation )

Returns: the similarity group of a canonical symplectic polar space

The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `Sp`, a GAP command returning the appropriate matrix group. Internally, the invariant form is asked, and the base change to our canonical form is obtained using the package form

    gap> GSpdesargues(4,GF(9));
    PGSp(4,9)

##### B.2-8 GammaSp

`‣ GammaSp`( d, f ) | ( operation )

Returns: the collineation group of a canonical symplectic polar space

The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `GeneralSymplecticGroup`, and adds the frobenius automorphism.

    gap> GammaSp(4,GF(9));
    PGammaSp(4,9)

##### B.2-9 DeltaOminus

`‣ DeltaOminus`( d, f ) | ( operation )

Returns: the similarity group of a canonical elliptic orthogonal polar space

The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `GOdesargues`, and computes the generators to be added.

    gap> DeltaOminus(6,GF(7));
    PDeltaO-(6,7)

##### B.2-10 DeltaOplus

`‣ DeltaOplus`( d, f ) | ( operation )

Returns: the similarity group of a canonical hyperbolic orthogonal polar space

The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `GOdesargues`, and computes the generators to be added.

    gap> DeltaOplus(8,GF(7));
    PDeltaO+(8,7)

##### B.2-11 GammaOminus

`‣ GammaOminus`( d, f ) | ( operation )

Returns: the collineation group of a canonical elliptic orthogonal polar space

The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `DeltaOminus`, and computes the generators to be added.

    gap> GammaOminus(4,GF(25));
    PGammaO-(4,25)

##### B.2-12 GammaO

`‣ GammaO`( d, f ) | ( operation )

Returns: the collineation group of a canonical parabolic orthogonal polar space

The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `GO`, a GAP command returning the appropriate matrix group. Internally, the invariant form is asked, and the base change to our canonical form is obtained using the package form. Furthermore, the generators to be added are computed.

    gap> GammaO(5,GF(49));
    PGammaO(5,49)

##### B.2-13 GammaOplus

`‣ GammaOplus`( d, f ) | ( operation )

Returns: the collineation group of a canonical hyperbolic orthogonal polar space

The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `DeltaOplus`, and computes the generators to be added.

    gap> GammaOplus(6,GF(64));
    PGammaO+(6,64)

##### B.2-14 GammaU

`‣ GammaU`( d, f ) | ( operation )

Returns: the collineation group of a canonical hermitian variety

The argument d is the dimension of the underlying vector space, f is the finite field. The method relies on `GU`, a GAP command returning the appropriate matrix group. Internally, the invariant form is asked, and the base change to our canonical form is obtained using the package form. Furthermore, the generators to be added are computed.

    gap> GammaU(4,GF(81));
    PGammaU(4,9^2)

##### B.2-15 G2fining

`‣ G2fining`( d, f ) | ( operation )

Returns: the Chevalley group G_2(q)

This group is the group of projectivities stabilising the split Cayley hexagon embedded in the parabolic quadric Q(6,q): X0X4+X1X5+X2X6=X32. f must be a finite field and d must be 5 or 6. When d is 5, F must be a field of even order, and then the returned group consists of projectivities of W(5,q). The generators of this group are described explicitly in [VM98], Appendix D. A correction can be found in [Off00]. However, also this source contains a mistake.

##### B.2-16 3D4fining

`‣ 3D4fining`( f ) | ( operation )

Returns: the Chevalley group 3D4(q)

The argument f must be a field of order q3 This group is the group of collineations stabilising the twisted triality hexagon embedded in the hyperbolic quadric Q+(7,q): X0X4+X1X5+X2X6+X3X7 The generators of this group are described explicitly in [VM98], Appendix D.

#### B.3 Basis of the collineation groups

The GenSS uses a function `FindBasePointCandidates` taking a group as one of the arguments. From a geometrical point of view, it is straightforward to construct a basis for a collineation group for the action on projective points.

##### B.3-1 FindBasePointCandidates

`‣ FindBasePointCandidates`( g, opt, i, parentS ) | ( operation )

Returns: a record

The returned record contains the base points for the action, and some other fields. The information in the other fields is determined from the arguments opt and i. More information on these details can be found in the manual of GenSS.

Variations on this version of `BasePointCandidates` are found in FinInG used in previous versions of GenSS. These variations are already or will become obsolete in the (near) future.

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