50.40 DnLattice

DnLattice( tbl, grammat, reducibles )

tries to find sublattices isomorphic to root lattices of type D_n (for n geq 5 or n = 4) in a lattice that is generated by the norm 2 characters reducibles, which must be characters of the table tbl. grammat must be the matrix of scalar products of reducibles, i.e., the Gram matrix of the lattice.

DnLattice is able to find irreducible characters if there is a lattice with n>4. In the case n = 4 DnLattice only in some cases finds irreducibles.

DnLattice returns a record with components

irreducibles:

the list of found irreducible characters,

remainders:

the list of remaining reducible characters, and

gram:

the Gram matrix of the characters in remainders.

The remaining reducible characters are transformed into a normalized form, so that the lattice-structure is cleared up for further treatment. So DnLattice might be useful even if it fails to find irreducible characters.

    gap> tbl:= CharTable( "Symmetric", 4 );;
    gap> y1:=[ [ 2, 0, 2, 2, 0 ], [ 4, 0, 0, 1, 2 ], [ 5, -1, 1, -1, 1 ],
    >          [ -1, 1, 3, -1, -1 ] ];;
    gap> g1:= MatScalarProducts( tbl, y1, y1 );
    [ [ 2, 1, 0, 0 ], [ 1, 2, 1, -1 ], [ 0, 1, 2, 0 ], [ 0, -1, 0, 2 ] ]
    gap> e:= DnLattice( tbl, g1, y1 );
    rec(
      gram := [  ],
      remainders := [  ],
      irreducibles :=
       [ [ 2, 0, 2, -1, 0 ], [ 1, -1, 1, 1, -1 ], [ 1, 1, 1, 1, 1 ],
          [ 3, -1, -1, 0, 1 ] ] )

DnLatticeIterative( tbl, arec )

was made for iterative use of DnLattice. arec must be either a list of characters of the table tbl, or a record with components

remainders:

a list of characters of the character table tbl, and

norms:

the norms of the characters in remainders,

e.g., a record returned by LLL LLL. DnLatticeIterative will select the characters of norm 2, call DnLattice, reduce the characters with found irreducibles, call DnLattice for the remaining characters, and so on, until no new irreducibles are found.

DnLatticeIterative returns (like LLL LLL) a record with components

irreducibles:

the list of found irreducible characters,

remainders:

the list of remaining reducible characters, and

norms:

the list of norms of the characters in remainders.

    gap> tbl:= CharTable( "Symmetric", 4 );;
    gap> y1:= [ [ 2, 0, 2, 2, 0 ], [ 4, 0, 0, 1, 2 ],
    >   [ 5, -1, 1, -1, 1 ], [ -1, 1, 3, -1, -1 ], [ 6, -2, 2, 0, 0 ] ];;
    gap> DnLatticeIterative( tbl, y1);
    rec(
      irreducibles :=
       [ [ 2, 0, 2, -1, 0 ], [ 1, -1, 1, 1, -1 ], [ 1, 1, 1, 1, 1 ],
          [ 3, -1, -1, 0, 1 ] ],
      remainders := [  ],
      norms := [  ] )

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