50.38 Extract

Extract( tbl, reducibles, grammat )
Extract( tbl, reducibles, grammat, missing )

tries to find irreducible characters by drawing conclusions out of a given matrix grammat of scalar products of the reducible characters in the list reducibles, which are characters of the table tbl. Extract uses combinatorial and backtrack means.

Note: Extract works only with ordinary characters!

missing:

number of characters missing to complete the tbl perhaps Extract may be accelerated by the specification of missing.

Extract returns a record extr with components solution and choice where solution is a list [ M_1, ldots, M_n ] of decomposition matrices that satisfy the equation [ M_i^tr cdot X = Sublist( reducibles, extr.choice[i] ) , ] for a matrix X of irreducible characters, and choice is a list of length n whose entries are lists of indices.

So each column stands for one of the reducible input characters, and each row stands for an irreducible character. You can use Decreased Decreased to examine the solution for computable irreducibles.

    gap> s4 := CharTable( "Symmetric", 4 );;
    gap> y := [ [ 5, 1, 5, 2, 1 ], [ 2, 0, 2, 2, 0 ], [ 3, -1, 3, 0, -1 ],
    >  [ 6, 0, -2, 0, 0 ], [ 4, 0, 0, 1, 2 ] ];;
    gap> g := MatScalarProducts( s4, y, y );
    [ [ 6, 3, 2, 0, 2 ], [ 3, 2, 1, 0, 1 ], [ 2, 1, 2, 0, 0 ],
      [ 0, 0, 0, 2, 1 ], [ 2, 1, 0, 1, 2 ] ]
    gap> e:= Extract( s4, y, g, 5 );
    rec(
      solution :=
       [ [ [ 1, 1, 0, 0, 2 ], [ 1, 0, 1, 0, 1 ], [ 0, 1, 0, 1, 0 ],
              [ 0, 0, 1, 0, 1 ], [ 0, 0, 0, 1, 0 ] ] ],
      choice := [ [ 2, 5, 3, 4, 1 ] ] )
    # continued in 'Decreased' ( see "Decreased" )

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