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10 Realization spaces of geometric lattices

The computation of the realization space uses the singular package to call functions from Singular [DGPS24]. Our implementation is an adaptation of the algorithm described in [Cun22, Algorithm 3.9] for geometric lattices of arbitrary rank.

10.1 Construction

10.1-1 LRealizationSpace

Returns: RealizationSpaceOfGeometricLattice

Computes the realization space of the geometric lattice L in characteristic char. Optionally, a generating set of L (as a subset of LAtoms(L)) can be given. Otherwise, the algorithm tries to find a small one.

gap> L:=IntersectionLattice(AGpql(3,3,3));
<Geometric lattice: 9 atoms, rank 3>
gap> RS:=LRealizationSpace(L,0);
<RealizationSpaceOfGeomLattice: in characteristic 0, non-empty: true>
gap> S:=LGenSet(L);
[ 2, 3, 4, 6, 8 ]
gap> RS:=LRealizationSpace(L,0,S);
<RealizationSpaceOfGeomLattice: in characteristic 0, non-empty: true

10.2 Attributes

10.2-1 LGenSet

Returns: a set

Computes some generating set of L, see [Cun22, Def.3.4] resp. [GMRS26, Def.4.2].

gap> L:=IntersectionLattice(AGpql(3,3,3));
<Geometric lattice: 9 atoms, rank 3>
gap> LGenSet(L);
[ 2, 3, 4, 6, 8 ]

10.2-2 RSLattice

Returns: a geometric lattice

Returns the geometric lattice of the realization space RS.

gap> RSLattice(RS);
<Geometric lattice: 9 atoms, rank 3>

10.2-3 RSCharacteristic

Returns: 0 or a prime number

Returns the characteristic of the reliation space RS.

gap> RSCharacteristic(RS);
0

10.2-4 RSDefField

Returns: a field

Returns the field over which the realization space RS is defined.

gap> RSDefField(RS);
Rationals

10.2-5 RSCoeffMat

Returns: a matrix

Returns the coefficient matrix of RS with entries in RSPRing(RS).

gap> RSCoeffMat(RS);
[ [ a1-1, a1, 0 ], [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ], 
  [ 1, 1, a1 ], [ 1, 1, 1 ], [ 0, 1, -a1^2+a1 ], 
  [ 1, 0, a1 ], [ -a1+1, -a1, -a1^2 ] ]

10.2-6 RSDimension

Returns: a non-negative integer

Returns the dimension of RS.

gap> RSDimension(RS);
0

10.2-7 RSPRing

Returns: a polynomial ring

The polynomial ring over which the representing coefficient matrix RSCoeffMat(RS) of RS is defined.

gap> RSPRing(RS);
Rationals[a1]

10.2-8 RSIdealMinors

Returns: an ideal

Returns the ideal of equations which have to satisfied due to the dependcy of atoms in RSLattice(RS) over RSDefField(RS).

gap> I:=RSIdealMinors(RS); GeneratorsOfIdeal(I);
<two-sided ideal in Rationals[a1], (1 generator)>
[ a1^2-a1+1 ]

10.2-9 RSNonMinors

Returns: a list of polynomials

Gives a list of polynomials which should not vanish due to independent subsets of atoms in RSLattice(RS) over RSDefField(RS).

gap> RSNonMinors(RS);
[ a1^3-a1^2, a1^3, -a1^3+2*a1^2-a1, -2*a1^2+2*a1-1 ]

10.2-10 RSEvalPoint

10.3 Properties

10.3-1 RSIsNonEmpty

Returns: true or false

10.4 Further (auxillary) Operations

10.4-1 LSubsetGeneratedByS

10.4-2 LIsGenSet

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