Lattices with Group Action
- Introduction
- Automorphism Group and Isometry Testing
AutomorphismGroup(L): Lat → GrpMatProperAutomorphismGroup(L): Lat → GrpMatAutomorphismGroup(L, F): Lat, [ AlgMatElt ] → GrpMatAutomorphismGroup(L, F): Lat, { AlgMatElt } → GrpMatAutomorphismGroup(F): [ AlgMatElt ] → GrpMat- Example: Automorphisms of E8 Acting on Vectors
- Example: Automorphism Group of a 19-Dimensional Lattice
IsIsometric(L, M): Lat, Lat → BoolElt, AlgMatEltIsIsomorphic(L, M): Lat, Lat → BoolElt, AlgMatEltIsProperlyIsometric(L, M): Lat, Lat → BoolElt, AlgMatEltIsIsometric(L, F₁, M, F₂): Lat, [ AlgMatElt ], Lat, [ AlgMatElt ] → BoolElt, AlgMatEltIsIsomorphic(L, F₁, M, F₂): Lat, [ AlgMatElt ], Lat, [ AlgMatElt ] → BoolElt, AlgMatEltIsIsometric(L, M): Lat, Lat → BoolElt, AlgMatEltIsIsomorphic(L, M): Lat, Lat → BoolElt, AlgMatEltIsIsometric(F₁, F₂): [ AlgMatElt ], [ AlgMatElt ] → BoolElt, AlgMatEltIsIsomorphic(F₁, F₂): [ AlgMatElt ], [ AlgMatElt ] → BoolElt, AlgMatElt- Example: Isometry of Barnes-Wall Lattices
- Automorphism Group and Isometry Testing over \({\mathbb{F}}_q[t]\)
DominantDiagonalForm(X): Mtrx[RngUPol] → Mtrx, Mtrx, GrpMat, FldFin- Example: Isometry Testing over a Polynomial Ring
AutomorphismGroup(G): Mtrx[RngUPol] → GrpMat, FldFinIsIsometric(G1, G2): Mtrx[RngUPol], Mtrx[RngUPol] → BoolElt, Mtrx, FldFinShortestVectors(G): Mtrx[RngUPol] → SeqEnumShortVectors(G, B): Mtrx[RngUPol], RngIntElt → SeqEnum
- Lattices from Matrix Groups
- Creation of \(G\)-Lattices
- Operations on \(G\)-Lattices
- Invariant Forms
- Endomorphisms
- \(G\)-invariant Sublattices
Sublattices(G, Q): GrpMat, [ RngIntElt ] → [ Lat ], BoolEltSublattices(G, Q): [ Mtrx ], [ RngIntElt ] → [ Lat ], BoolEltSublattices(L, Q): Lat, [ RngIntElt ] → [ Lat ], BoolEltSublattices(G, p): GrpMat, RngIntElt → [ Lat ], BoolEltSublattices(L, p): Lat, RngIntElt → [ Lat ], BoolEltSublattices(G): GrpMat → [ Lat ], BoolEltSublattices(L): Lat → [ Lat ], BoolEltSublatticeClasses(G): GrpMat → [ Lat ]- Example: Sublattices of a G-Lattice
- Example: Sublattices with Projections
- Lattice of Sublattices
- Creating the Lattice of Sublattices
SublatticeLattice(G, Q): GrpMat, [ RngIntElt ] → LatLat, BoolEltSublatticeLattice(G, Q): [ Mtrx ], [ RngIntElt ] → LatLat, BoolEltSublatticeLattice(G, p): GrpMat, RngIntElt → LatLat, BoolEltSublatticeLattice(G, p): [ Mtrx ], RngIntElt → LatLat, BoolEltSublatticeLattice(G): GrpMat → LatLat, BoolElt- Example: Creating a Lattice of Sublattices
- Operations on the Lattice of Sublattices
# V: LatLat → RngIntEltV ! i: LatLat, RngIntElt → LatLatEltV ! M: LatLat, Lat → LatLatEltV ! M: LatLat, Mtrx → LatLatEltNumberOfLevels( V ): LatLat → RngIntEltLevel(V, i): LatLat, RngIntElt → [ LatLatElt ]Levels(v): LatLat → [ [LatLatElt] ]Primes(V): LatLat → [ RngIntElt ]Constituents(V): LatLat → SeqEnumIntegerRing() ! e: RngInt, LatLatElt → RngIntElte + f: LatLatElt, LatLatElt → LatLatElte meet f: LatLatElt, LatLatElt → LatLatElte eq f: LatLatElt, LatLatElt → BoolEltMaximalSublattices(e): LatLatElt → [ LatLatElt ], [ RngIntElt ]MinimalSuperlattices(e): LatLatElt → [ LatLatElt ], [ RngIntElt ]Lattice(e): SubModLatElt → LatBasisMatrix(e): SubModLatElt → MtrxMorphism(e): SubModLatElt → Mtrx- Example: Invariant Lattices of the Root Lattice A5
- Example: Lattices Invariant under SL(2,7)
- Bibliography