Example GrpPC_Extensions (H70E51)

> G := RandomGenus2Group(3^2, [6]);
> t := pCentralTensor(G, 1, 1);
> t;
Tensor of valence 3, U2 x U1 >-> U0
U2 : Full Vector space of degree 12 over GF(3)
U1 : Full Vector space of degree 12 over GF(3)
U0 : Full Vector space of degree 4 over GF(3)
> Genus(t);
2    

The Pfaffian of t is an irreducible cubic over GF(9), so computing its pseudo-isometry group may contain elements of (Gal)(GF(9)).

> PI := TGPseudoIsometryGroup(t);
> Random(PI);
[0 1 0 0 0 2 1 1 0 2 1 0 0 0 0 0]
[2 0 0 0 1 0 0 2 1 0 1 2 0 0 0 0]
[1 1 0 0 0 1 1 1 0 0 2 1 0 0 0 0]
[0 2 0 0 2 0 0 2 0 0 1 1 0 0 0 0]
[1 2 2 0 0 2 2 2 1 0 0 0 0 0 0 0]
[2 2 2 1 1 0 0 1 1 2 0 0 0 0 0 0]
[2 2 2 0 1 0 1 0 2 2 1 2 0 0 0 0]
[0 1 2 1 1 2 1 2 0 1 2 2 0 0 0 0]
[2 0 1 0 2 2 2 1 1 1 0 0 0 0 0 0]
[2 1 1 2 0 1 1 1 0 2 0 0 0 0 0 0]
[0 2 0 1 2 2 1 1 0 1 1 0 0 0 0 0]
[1 0 2 0 0 1 0 2 2 0 1 2 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 2]
[0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0]
> Factorization(#PI);
[ <2, 8>, <3, 7>, <5, 1>, <7, 1>, <13, 1>, <73, 1> ]

Indeed, the pseudo-isometry group may not be linear over the centroid but semi-linear. To see the discrepancy, we rewrite t over its centroid and determine its pseudo-isometry group as a subgroup of GL_E(V) x GL_E(W). Note that this group is smaller by a factor of 2. We see that the pseudo-isometry group of t is strictly semi-linear over GF(9), as there exists a Galois involution.

> s := TensorOverCentroid(t);
> s;
Tensor of valence 3, U2 x U1 >-> U0
U2 : Full Vector space of degree 6 over GF(3^2)
U1 : Full Vector space of degree 6 over GF(3^2)
U0 : Full Vector space of degree 2 over GF(3^2)
> PI_C := TGPseudoIsometryGroup(s);
> Random(PI_C);
[    1 $.1^7     1 $.1^2 $.1^2 $.1^5     0     0]
[  $.1     2 $.1^7 $.1^3 $.1^6 $.1^5     0     0]
[$.1^7 $.1^2 $.1^7     1 $.1^6 $.1^2     0     0]
[$.1^7 $.1^7 $.1^7 $.1^3 $.1^7     1     0     0]
[    2 $.1^6 $.1^3 $.1^2 $.1^3     0     0     0]
[$.1^7   $.1 $.1^7 $.1^6 $.1^6 $.1^7     0     0]
[    0     0     0     0     0     0     2 $.1^7]
[    0     0     0     0     0     0 $.1^3 $.1^5]
> Factorization(#PI_C);
[ <2, 7>, <3, 7>, <5, 1>, <7, 1>, <13, 1>, <73, 1> ]

Canonical Labelling

TGSignature(G) : GrpPC -> List

Given an odd-ordered p-group G isomorphic to A x H of genus ≤2, class ≤2, and exponent p, return the canonical tame genus signature as a list with four entries. For groups with genus ≤1, the last two entries will be empty. We describe the entries.

(1)

The first is a triple of three integers (q, d, e) such that associated fully nondegenerate commutator tensor of H has the form GF(q)^d x GF(q)^d ↣ GF(q)^e. Here, GF(q) is the largest field F which the commutator tensor is F-bilinear.

(2)

The second entry is a pair of integers (r, c), where A isomorphic to (Z/(p))^r and c=0.

(3)

The third entry is a sequence of odd integers corresponding to the dimensions of the flat indecomposable spaces.

(4)

The fourth and last entry is a list sequences in GF(q) of possibly different lengths. A sequence [c₁, ..., c_{d + 1}] in the second entry corresponds to the coefficients of the homogeneous polynomial in x and y in degree d:

c₁x^d + c₂ x^{d - 1}y + c₃ x^{d - 2}y² + ... + c_{d + 1}y^d.

Because this is a canonical label, two groups are isomorphic if, and only if, their tame genus signatures are equal. See [BMW17] for details on how the last two entries describe a genus 2 group.

TGSignature(t) : TenSpcElt -> List

Given an alternating, directly indecomposable tensor t:V x V ↣ W of genus ≤2, where V and W are GF(q)-vector spaces for odd q, return the canonical tame genus signature as a list with four entries. For tensors with genus ≤1, the last two entries will be empty. We describe the entries.

(1)

The first is a triple of three integers (q^f, d, e) such that associated fully nondegenerate tensor of t has the form GF(q^f)^d x GF(q^f)^d ↣ GF(q^f)^e. Here, GF(q^f) is the largest field F which the associated fully nondegenerate tensor is F-bilinear.

(2)

The second entry is a pair of integers (r, c), where the radical of t is r-dimensional and the codimension of the image of t in W is c, both are determined over GF(q).

(3)

The third entry is a sequence of odd integers corresponding to the dimensions of the flat indecomposable spaces.

(4)

The fourth and last entry is a list sequences in GF(q) of possibly different lengths. A sequence [c₁, ..., c_{d + 1}] in the second entry corresponds to the coefficients of the homogeneous polynomial in x and y in degree d:

c₁x^d + c₂ x^{d - 1}y + c₃ x^{d - 2}y² + ... + c_{d + 1}y^d.

Because this is a canonical label, two tensors are pseudo-isometric if, and only if, their tame genus signatures are equal. See [BMW17] for details on how the last two entries describe a genus 2 tensor.

> blocks := [1, 1, 1, 2, 2, 3, 3, 5, 6];
> G := RandomGenus2Group(9, blocks);
> #G eq 9^(&+blocks + 2);
true    

> TGSignature(G);
[* <9, 21, 2>, <6, 0>,
    [ 3, 3, 5 ],
    [*
        [ 0, 1 ],
        [ 1, 1 ],
        [ 1, 2, 2, 2 ]
    *]
*]    

> TG_SG := SmallGroupProcess(3^7, IsTameGenusGroup);
> 
> repeat
>     G := Current(TG_SG);
>     _, ID := CurrentLabel(TG_SG);
>     print "Small group ID:", ID;
>     TGSignature(G);
>     Advance(~TG_SG);
> until IsEmpty(TG_SG);    

Small group ID: 9106
[* <3, 3, 2>, <2, 0>,
    [ 3 ],
    [* *]
*]
Small group ID: 9107
[* <3, 4, 2>, <1, 0>,
    [],
    [*
        [ 1, 0 ],
        [ 0, 1 ]
    *]
*]
Small group ID: 9108
[* <3, 4, 2>, <1, 0>,
    [],
    [*
        [ 0, 0, 1 ]
    *]
*]
Small group ID: 9109
[* <9, 2, 1>, <1, 0>, [* *], [* *] *]
Small group ID: 9110
[* <3, 5, 2>, <0, 0>,
    [ 3 ],
    [*
        [ 0, 1 ]
    *]
*]
Small group ID: 9111
[* <3, 5, 2>, <0, 0>,
    [ 5 ],
    [* *]
*]
Small group ID: 9302
[* <3, 2, 1>, <4, 0>, [* *], [* *] *]
Small group ID: 9305
[* <3, 4, 1>, <2, 0>, [* *], [* *] *]
Small group ID: 9308
[* <3, 6, 1>, <0, 0>, [* *], [* *] *]
Small group ID: 9310
[* <3, 7, 0>, <7, 0>, [* *], [* *] *]